This is my jungling Amumu guide that I have lead my team to late game ace's for surrenders and good games if he is build right, Amumu is one of the most effective champions in the game. I'll be explaining how I play Amumu and how he can be very effective in team fights and ganking.

Pros
-Very Effective Late Game with a game changing ultimate.
-With Blue buff can spam Tantrum wiping small camps out very fast
-Good initiations with Bandage Toss for ganking and escaping through the jungle
-Great with AoE champions with his Curse of the Sad Mummy



Cons
-Very Mana Dependant early game without blue leading to high chances of invasion
-Slower than other junglers
-Need to be able to hit his Bandage Toss or ganks are failed
-Doesn't bring positive attitude to the team

is Amumu's passive, and it's what makes him so dangerous to many champions. At level 18, you lower magic resistance by a full 35 points which in champion's MR gets reduced to zero if they haven't gotten any Magic Resist items. This makes your magic DoT and your ultimate that much more devastating.
is Amumu's skill shot that has a pretty long range and is really great for initiating. You shoot out a bandage in a straight line which stuns the first enemy champion or minion that it hits. You also deal magic damage on impact and you pull yourself to them
This ability is what makes Amumu's jungle possible. It is an ability that creates an AOE pool that clears the jungle without any problems while spamming ur Tantrum.
is your spamming spell that is reduced by half a second for every hit you take as it's recharging which leads your main spell of farming. It hits all enemies around you for magic damage. Passively, it reduces all physical damage you take by amounts increasing up to 5. This is very effective in mid game and makes you very survivable in the jungle.
is your game winning ultimate. You turn a pretty large area around you into the a circle of destruction for ur team to fully engage onto the enemy without them being able to defend for 2 seconds. The initiation process should preferably start by tossing ur Bandage Toss and then casting your Curse of the Sad Mummy

Greater Mark of Magic Penetration
I take 9 magic pens for more effective ganks early
Greater Glyph of Scaling Magic Resist
Magic resistance is a must have for any tanks. enough said
Greater seal of resilence
Same for armour seals, its a must have.
Greater Quintessence of Health
I take this because it allows me to finish my jungle route and have enough health for a gank at top or mid before going back to buy.
Greater Quintessence of Attack Speed
Is another good one to clear the jungle faster but you don't get enough sustain as you do with +78 health.

For my first items i take Regrowth Pendant and 1 Health Potion and the route i run is wolves u start with your Despair and get some help from your mid for faster kills, then move to ancient golem make sure u get a leash and as soon as u can Smite then grab your Tantrum and finish the camp then move onto wraiths return to wolves and grab your Tantrum again then go to lizard elder and use your Health Potion and Smite then either wraiths or golems depending on whether bot lane needs a gank or mid, of course then grab your Bandage Toss at 4. Then after u finished your route and checked for ganks if none kill wraiths, golems, wolves again then you should be level 5 and have more 900 which is enough to get 2 sight ward your philosopher's stone Boots 2 Health Potion and 1 or 2 Mana Potion

Smite just the must have in the jungle and it's 10g every smite you use so never be afraid to use it if buffs and dragon or baron are on respawn for extra gold.
Flash In my opinion there's no better way to initiate team fights than to Flash amd use Curse of the Sad Mummy
Teleport As Amumu, you are great at killing minions with your aoe damage, and Tantrum will be spammed when you get hit by the minions. Overall Teleport is great for pushing and for defending. but I wouldn't advice it when jungling.
Ghost Ghost isn't completely useless when using Amumu I prefer flashing but ghost is good for when ur team doesn't have someone that can chase and keep up with the opposing champions.

Mobility Boots I grab these if none of the lanes are fed and if you plan on ganking a lot because there's nothing more annoying than a Amumu out running you with Despair and Tantrum slowly killing you.
Ninja Tabi These are really useful to take if the team has 3 AD champions.
Mercury's Treads I usually take these anytime the team has 2 AP champions, the MR is really nice and the 35 tenacity is always nice.
Boots of Swiftness These are good boots to take to allow you to have the constant +3 enhanced movement speed.
Glacial Shroud With 15% CDR, 425 extra mana and 45 armor you essentially become a skill spammer with your Tantrum, also reducing the cooldown on your Curse of the Sad Mummy
Force of Nature In my opinion a must have for Amumu, the +76 magic resistance is always good to have plus that +40 health regeneration with the passive that restores 1.75% of your maximum health.
Frozen Heart A must have with Amumu the 99 armor and 500 mana allows you to be able not have to worry about running out of mana, plus the passive slows attack speed of enemy champions attacks by 20%.
Sunfire Aegis This is a strong item for Amumu the +450 health and +45 armor is always helpful to slow the AD carry from tearing thru your team. Plus the passive will speed jungling up greatly.
Randuin's Omen A very strong item with it's +75 armor and +350 health with a really strong active. You convert your Heart of Gold into this to obtain a strong tanking item with a powerful active. When initiating with your Bandage Toss into the enemy team then use your Curse of the Sad Mummy puts most of the enemy team unable to defend for about 2 seconds then you activate your Randuin's active, you can continue to prevent them from really doing any damage alone with your Frozen Heart.
Abyssal Mask Always a good item to grab for your last item it gives you +70 ability power plus +57 magic resistance, with a good passive that reduces all enemy champions magic resistance near by 20 so your AP carry can do more damage to there team.
Banshee's Veil This is not a must have item for Amumu but if the enemy team has a lot of magic damage it is great for it's +50 Magic Resistance and the passive allows you to block spells every 45 seconds. Also never hurts to have more mana and health.
Guardian Angel This is a good item to grab if you're getting focused in team fights and dying too fast, however you shouldn't be you're the tank.
Warmog's Armor I rarely ever get this item when playing Amumu but it gives tons of health and has a really good health regeneration to go great with your Force of Nature. I would only advice thinking about picking this item up if you already have +200 armor and magic resist against a non-fed team.

Buying and Placing Sight Wards are a very important role of the jungle to make sure your team is always in vision of the enemy jungler and knowing when there is a lane missing and where they are heading. You also need to grab an Oracle's Elixir to clear there vision of when you're ganking them.