When playing Amumu jungle there is one very important thing and that is to land your Bandage Toss. Bandage Toss is your engage when ganking, unless you lanegank. So if you’re not good with skillshots and can’t predict your opponents movements, then there might be some problems playing Amumu.
I have been playing Amumu since i started playing LoL 3 years ago and I have a bit over 2k games right now. And even though so many cool new champions came out the past years he is still by far my favorite champion, therefore I think that I have a good amount of experience with him and a bunch of different builds, though i will mainly focus on the build I find best in the current jungle and meta. There will be a brief introduction to other items that is good on Amumu but not a whole chapter.

The runes used for Amumu can be different from when you want to build tanky and when you want to deal a lot of damage, there is also different kind of setups for the damage runes and the tanky runes:

The tanky set of runes is with armor in Marks and Seals which does it so Amumu wont take as much damage in the jungle and against the ranged ad carry nor top. The Glyphs are scaling Magic Resistance because you want to get tankier and be able to stand in the face of the enemy AP carry without dying instantly. The last thing is the Quints which is Movement Speed. The reason i've picked the MS Quints is because i like running a bit faster than my opponent while my Despair is on, also the bonus MS might do so i can be faster in the lane for a gank.

These runes are more for the person that wants to deal a lot of damage and clear the jungle faster. Though there might be a downside to these runes and that is that you take way more damage than you would with the other runes. The Marks are Magic Penetration since Amumu is a Magic damage based champion, so the Magic Penetration is the best if you want to rely more on damaging. The Seals are armor so that you can still survive in the jungle, since the camps now deal more damage than in the season 2 jungle. The Glyphs are scaling Ability Power, since your abilities scale of Ability power instead of Attack Damage. The Quints are Magic Penetration for the same reason as the Marks, though they can be changed for flat ability power if you feel that it suits you better.

I go with 0/21/9 with masteries. It looks like the normal defensive tree for a jungler though there is some personal choices like Reinforced Armor and Juggernaut . I take Reinforced Armor due to the fact that Im the teams tank and therefore I should be able to let the enemy ranged ad carry shoot at me. Lategame that is easily around 850 damage. With Reinforced Armor the 850 damage is decreased by 85, which then is 765. So you can handle a lot more shots than if you didn’t have it in your masteries. Juggernaut is chosen over masteries like Legendary Armor , Relentless and Resistance because:

• Legendary Armor :

This mastery is great with very high armor champions like Malphite, since Malphite will get around 400 armor he will get additional 20 from the mastery. But with the build I made for Amumu he will only get around 8 armor.

• Relentless :

Relentless is alright if you don’t have anything else to go up in for masteries, but since I already have 15% tenacity from Tenacious and will get another 35% from Spirit of the Ancient Golem, I don’t see the slow as a problem. I do know that there’s a difference to reducing the slow and reducing the time of the slow, but I still think tenacity is a better idea.

• Resistance :

I like the fact of getting Magic Resistance so that you can early gank mid, but if I have to choose between getting 4% additional health and getting 3 Magic Resistance(there’s already 1 point in it) then I’d rather get the health.

Reason for taking masteries:

• Summoner's Resolve : This masterypoint is normal for a jungler since you will get more gold from using smite, and it's 1 point well spent.
• Durability : You want to be tanky and you need health, also you want this mastery so that you can get a point in Veteran's Scars .
• Tough Skin : Once again you want to be tanky and take as little damage as possible, this helps you out much in the jungle during the low lvl's with the passive from your Tantrum
  Hardiness : Most of the damage in the jungle is physical damage and so is the basic attacks from champions. So getting additional armor is a must. The tankier the better
• Resistance : There is also Magic damage in the junlger( Baron Nashor deals magic damage)and in the form of Magic damage based abilities from champions.
• Bladed Armor : When you get tanky and is not a damage bomb, then you might wanna deal the damage you can, so dealing 6 damage every time a monster in the jungle is hitting you is a bonus.
• Veteran's Scars : To start out with additional 30 HP is great at it might be the 30 hp that you have left after a gank!
• Tenacious : As a tank and initiator you're gonna get in the fight and take the abilities face on. With the extra 15% and 35% from Spirit of the Ancient Golem you get 50%, therefore only taking half of the Crowd Control.
• Juggernaut : As a tank i love having much health, and with Juggernaut i get more, nothing much to argue with there!
• Reinforced Armor : Like i explained earlier in the chapter it makes you tank more since a lategame Ranged AD Carry can critical hit for 850 damage it will reduce the damage to 765 so you will be able to tank for a longer time.
• Honor Guard : Tank ALL the damages!
• Summoner's Insight : This will decrease the time for your flash which might be crucial for survival, so it's a pretty good idea to spend a masterypoint in it.
• Wanderer : Instead of using 3 masterypoints in Meditation i like to put it in Wanderer since i have the opinion that if you need 3 mana regeneration, which is only 0.6 per second. If you really need it more than the increased movement speed then you spent your mana wrong.
• Improved Recall : You gonna get back faster! Might at well save some time.
• Mastermind : With Mastermind you will get your Flash and Smite faster which is pretty awesome and your gold income will increase since Smite gives you gold when used.
• Runic Affinity : Amumu is benefiting very much from the blue buff since he is a caster and his abilities is his damage and they cost mana. So it's great for him to extend the time he is having the blue buff.

The 21/9/0 masteries is for the person who wants to deal a load of damage and doesn't care much about being squishy. Amumu does have good scaling on his Despair and Tantrum, so he is playable as a AP jungler. He might be getting very squishy towards the lategame, but if you play well then you can pull it off.

Reason for taking masteries:

• Sorcery: Amumu is a caster and his cooldown on his ultimate Curse of the Sad Mummy is quite long, which is understandable since it's one of the best initiates for a teamfight. So if you can shorten the cooldown, then go for it!
• Butcher : The only reason i've taken a point in Butcher is due to the requirement of taking 16 points in the offensive tree if you want Archmage , which we do since we're going offensive. It is a possibility to take the point in Spellsword instead, but Butcher is worth more during the low lvl's in the jungle and it takes at least 40 AP for Spellsword to be better, which is much for a jungler that doesn't buy AP items untill midgame.
• Blast : If you ask me Blast is not that great at all, the reason to take it is the chained mastery Arcane Knowledge which offers you Magic Penetration.
• Havoc : Havoc is a good mastery the champions that will deal a lot of damage, which AP Amumu is.
• Arcane Knowledge : Once again, Amumu is a caster and does Magic damage, so the 8% Magic Penetration is a great asset to Amumu.
• Mental Force : We're focusing on a AP build, so having the flat AP from the start is good and makes you clear the jungle faster.
• Archmage : This is the mastery mentioned earlier. It increases the AP you got by 5% when rank 4. This means more damage and more damage is what we're aiming for!
• Executioner : I mainly get Executioner for assassins, but since Amumu does a lot of damage with his initiate they drop pretty fast and you bring them under 50% HP.
• Summoner's Resolve : This masterypoint is normal for a jungler since you will get more gold from using smite, and it's 1 point well spent.
• Durability : Even though you're getting offensive you still need to make sure that you don't die, with more health you wont.
• Hardiness : This gives you more armor which makes it easier for your jungling, since you won't take as much damage.
• Veteran's Scars : Like the Durability you get more health which is great for you since you wont be as squishy as you would without.

So i've heard a lot of people discussing if you should start with Despair instead of Tantrum and that it's better damage. But i always pick Tantrum since i find it better. I then decided to try and explaining it myself!

When you start with Tantrum you take 2 less physical damage and because of Tough Skin from the masteries you decrease the damage by 2 more, which is 4. If we count in the 48 armor you have at lvl 1 then you reduce the damage by a lot, also the skill itself does 75 damage of the skills cooldown is decreased when you're hit by a physical attack, which means that you can take 2x2 hits from the minor minions before they die and several from blue. Now if you start with Despair you deal a percentage (1.5%) of the golem’s maximum health but at lvl 1 the golem only has 1500 health which means you damage 22 + 8 flat damage to the golem every second. By experience i would say that Tantrum is better to start out with.
The damage is increased though, do to his passive Cursed Touch, which reduces the Ancient Golems Magic Resistance to -15, therefore the 75 damage is increased and so is the damage from Despair, by 15%. Tantrums damage is increased to 86.25 and Despairs damage is now 34.5 damage.

Also to try and come with a better reason than just saying “Well duh it’s better: v” I did a bit of math:

With 47.18 armor at lvl 1 Amumu reduces damage by 32%, also due to the passive on Amumu's Tantrum and the Masterypoints taken in Tough Skin the damage is reduced by another 4. The Ancient Golem does 65 damage and with the reduction from Tantrum, Tough Skin and Honor Guard you get a reduction of 41%.
If you start with Despair you have a damage reduction of the standard 32% and then the Honor Guard + Tough Skin which increases it to 37%. Though there’s a difference regarding flat reduction instead of percentage.

This might be slightly off since I’m not pro at math.

In my normal build i get Sorcerer's Shoes with Alacrity enchantment since Amumu's damage is magic damage and i don't need the Tenacity from Mercury's Treads since i'm building Spirit of the Ancient Golem which grants Tenacity.

Spirit of the Ancient Golem is build to make something out of Spirit Stone which is the other upgrade from Machete instead of Madred's Razor. Also Spirit of the Ancient Golem is a great tank item since it grants you Tenacity, health, mana regeneration, health regeneration and it has the passive that makes jungling way easier.

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Wolves > Ancient Golem(Prefer smiteless) > Wraiths > Lizards > Duo Golems > Wraiths
By this time you should be around lvl 4. If you want to gank early then you should gank top right after taking red and then go back and take the Duo Golems.

When taking blue there is quite a big difference for a jungler if t is taken smiteless(Top and mid damages the blue a lot). If you get your blue smiteless your cleartime if about 5 seconds faster, but that’s not what’s most important. What’s important is your health. If you smite your blue and then don’t smite red you end up with around 1/5 of your health after pinging lvl 4, but when it’s smiteless you end up at almost half health.

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I really like to counterjungle and i see a lot of people failing to do it when i see them play. If you want to counterjungle then remember to always put a ward in the opponents blue brush or red brush, depending on what buff you're taking.

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Amumu can be a heavy manaconsumer but after taking the second blue you should try to let your mid have the blue since they will benefit more from it than you. Though you might not need the second blue, it depends on if you counterjungled enemy blue. If you have it you might as well give your own midlaner your blue. Amumu does benefit from blue and it’s very positive if he gets it.
When you're getting up in lvl's you shouldn't use your Despair on small camps since you clean way faster.

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Until you get your Liandry's Torment you don’t deal extremely much damage except if you use your ulti, since your damage is overtime based damage. But when you get Liandry's Torment you rip a good amount of percentages from your opponent and therefore it's easier to towerdive, take baron etc. Though there might be some opponents that is easier to dive because of low hp, no escape/juke etc. so if you have enough hp then you can make successful ganks.

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Pro:
Fearsome initiate
Strong lategame
Fast cleartime

Cons:
Needs his early blue and is easily counterjungled
Your skillshot(bandage toss) is crucial to hit
CD on ultimate is very high

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