There are 4 bosses currently in the game.
Cyclops Den - You can punch him repeatedly in the stomach, although this is not a good tactic as you can uppercut until just past the apex of the jump and punch him in his weak spot, which with his stun can make him very easy to juggle (except for stairs going up where you can't reach his eye with an uppercut). His 2 moves are an uppercut that flings you, a Melee move that knocks you back; block to counter them. If you have Thunderpunch, you can punch him in the stomach with a solid punch and it will chain towards his eye.
It is also possible to move fast enough to escape him. By using "Shadowdodge, Big Dash" and dodge-dashing right as he punches, or by using "Air Wolf, Hyper Fighting" and gaining velocity after repeated Air Wolfs, you can outrun the Cyclops. Doing this counts as defeating the boss, awarding you the points.
CYBERclops - Jump over the fist and slam him in the eye (Or uppercut then down-ercut on his "brain tube" as gnomey calls it). When you hit his weakspot he will move back and do a wide swinging punch. His normal punches can be dodged easily if you stop by blocking, so a reliable tactic is to wait until he punches once and then attack.
Haunted Megaskull - Uppercut him in the jaw as much as possible, but don't attack if he is too high up, otherwise the uppercut will connect and not stun him, and he will bite you.. You can also jump up and punch him in the head. Only the weakspot is effective because punching him in the head doesn't stun him and you are more vulnerable to attack if you keep attacking from there.. If you land a good punch, his eyes will turn blue and he will vomit out little bouncing skulls that can hurt you. Punch the mini skulls into the bigger skull for more damage. If the miniskulls collide with each other, you can score some extra Punchos. Hitting the miniskulls is not entirely reliable, so don't do it if you're low on health.
Defeating a boss will reward the player with 10000pts along with one of the lines, "MUST BE A MINI-BOSS", "LIKE A BAUS" or "NOT THE BOSS OF YOU" (also gives 2 chests and or 1-2 money bags or 1-3 idols)
SPARTAN MODE ONLY!
Gnorb - Upon reaching 1500 punchos or going far enough in a single run of Spartan Mode, the next dungeon segment you come across will be called "BEHOLD THE GNORB", and no more enemies will spawn. The Gnorb will greet you with "IT'S-A ME, GNORBIO" and then he will speak with you when you encounter him, which is the hallucination gnome head inside an orb. He will commend you for making it this far and try to give you an "extremely tiny sack of punchos" but the Gnorb throws it past you by accident. Then he will talk to you using random lines like "Oh nuts" until you try to punch past his forcefield. You can only move forward using solid dashing punches until you make contact with his orb, where he will change into an angry looking gnome head and write in a flashing orange font, "YOU DARE?!". The Gnorb will fire a flashing gnome hat rocket at you, which you can either dodge by jumping at the last second or punch (cannot be blocked), where it will bounce back into the Gnorb and destroy his forcefield and stun him, where you can punch him until the gnome head inside is back to normal. Keep doing this until he is destroyed, where it will say "OW MY BALL" and award the player 25000 points along with the line, "WAS A BIT OF A JERK"