Hey there, Mobafire. This is my first guide, so if you're going to leave me some suggestions, go right ahead and do so, but try to be nice about it. The reason I'm publishing this right now, is because I'm seeing if all of the things work, and making sure I did it right. I will also be putting more detail into everything over time. Give me a break though, it's alot of typing.

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- The defender of tomorrow.

I play Jayce as a jungler. He can get through it at a good speed, and he doesn't lost too much health, either. I will be updating this guide over time, and doing everything I can to make it the very best, like no one ever was..

Armor Pen - You'll be killing the fu-- outta the enemy team. They're going to try to get some armor, eventually. Trust me. Also nice for going against tanks.
Armor - Sustainability in the jungle.
Attack Speed - Although you are getting blue first with my route, I get Attack Speed runes for the passive on his W. (Get a number of mana back each basic.)
Quintessence - More armor pen. You can either use that, or Flat AD quints. It's up to you, really.

I start off with Cloth armor and 5 health pots. After I clear my jungle route, and get 1 or 2 kills, and my lanes are ok, I recall getting my boots in one run. I then try to help my top out, or go back in the jungle. You can help a lane push a turret, or you can go into their jungle. It's up to you. Gank mid, bot, top, get a tower, get dragon. Just try to help your team, and communicate with them.

21-9. I get MORE armor pen, some life steal, base ad, and some nice armor for jungling.

Passive - Hextech Capacitor An amazing passive. Gives a movement speed boost when you activate your ult, and since your ult costs no mana.. Use this to your advantage when chasing champions down, or running away. Also speeds up jungling.

Q - To The Skies! / Shock Blast To the skies: Jumps to an enemy, dealing damage and slowing them. Shock blast: Jayce fires a ball that detonates by hitting an enemy OR by reaching the end of its path, and has a small AoE. To the skies is a great move, because it slows them. If your top lane is pushed all the way to it's turret, all you have to do is jump to them, and E them into turret's range. Shock Blast is the skill that's going to piss off your enemies. The fact that with your E you can shoot PAST turret without getting shot at or even targeted, makes this an amazing move.

W - Lightning Field / Hyper Charge Lightning field: You restore a certain amount of mana per basic with your hammer passively. When activated, you have an aura around you that damages enemies for a few seconds. Hyper Charge: For the next 3 basics, Jayce's attack speed goes to 2.5 and does a certain % of damage per shot, depending on level.

E - Thundering Blow / Acceleration Gate Thundering blow: You swing at them, knocking them back and dropping them based on the % of MAX health they have. Acceleration gate: You place a gate (about the same size wall as karthus at level 1) that makes you go faster when you walk through it. Also, if you shoot your q through it, it will have increased range, and it will deal more damage while going faster. Yeah. Pretty epic.

R - Mercury Cannon / Mercury Hammer You start off in Mercury hammer form. When you press R, it changes to mercury cannon. Your first basic in Mercury cannon form will lower their armor and magic resist. Switching back to mercury hammer causes your first basic in hammertime mode to deal a bonus amount of damage, and gives you BONUS armor and magic resist. Use these to your advantage. Maybe try switching to your cannon, lowering their armor, hyper charge, gate q, hammertime, q, knockback e, and then basic them or switch back to cannon mode. So. Many. Combos.

AND OF COURSE

Flash is a great move to take with jayce, and Smite for jungling.
Some other good options:

or

Just make sure to have smite.

Starting off with Cloth Armor and 5 health pots, I start running to blue. While running, switch to cannon.

At 1:55, blue will spawn. Make sure to basic with your cannon once, and then W. When blue hits you once, activate a health pot. Then, switching to Hammerform, basic, and W. Smite it, kill the minions.

Heading to wolves, switch to cannon. Q over the wall covering wolves, basic, w, health pot, hammer jump, w.

Switch to cannon to run to wraiths.

Q over the wall, and basic the big one, health pot and then activate your w. Hammer jump, W. By this time, smite should have 10 seconds or so left on cd. If it's longer, you may want to run to wolves and get them again.

For Red buff, switch to cannon, use your w, basic once, and then ult switching to hammer and then w and q. Switch to cannon again if you want, or smite it if it's low enough. Make sure to use your health pots. At this point, you can gank top or bot or mid or whatever, or go after golems.

Jayce has amazing gank potential. He can get to lanes fast, and he has quite the amount of skills to help you chase or run away. I usually start with a gank at top. I ping a champ, and then when my solo top says ok to me or responds in chat, we go for him.
Here's a combo for jayce.

CANNON --> GATE --> Q THROUGH GATE --> RUN THROUGH GATE AND W --> SWITCH TO CANNON ONCE W IS DONE --> (At this point the person should be dead if you both targeted them. If not, W and basics. Going after the next one, I Q to them, slowing them and then walk IN FRONT of them, and e them behind me. At this point, just kill him with your cannon or hammer. The combos are endless.

Will update soon