Are there any bodyweight bicep exercises? Or any form of bicep exercise that can be performed without equipment?
There are upper body movements that involve the biceps, but none that isolate them specifically like you can with dumbbells or machines. Examples include:
Basically anything that involves pulling will involve the biceps, and more so if you do it with a suppinated grip.
Chin-ups pull-ups pull-ups pull-ups chin-ups pull-ups, chin-ups chin-ups. Chin-ups or pull-ups, as well as chin-ups, or chin-ups and pull-ups, alternating between chin-ups and pull-ups on different days, chin-ups chin-ups chin-ups, pull-ups chin-ups, chins, pulls, chinning and pulling. Specifically, chin-ups and pull-ups.
Chin-up negatives, chin-up holds at the top, chin-up holds at the bottom, chin-up Frenchies, chin-up ladders, maximum repetitions of chin-ups, many many sets of just a few chin-ups, narrow grip chin-ups, wide grip chin-ups, alternated grip chin/pull-ups, towel chin-ups, pull-ups leaping to another bar, chin-ups with a slide to each side at the top, chin-ups with extra weight, weighted pull-ups, greasing the groove with chin-ups throughout the day, chin-ups with short rest periods, chin-ups with long rest periods, chin-ups in a circuit with dips, ring chin-ups, clapping pull-ups, muscle-ups, 50 chin-ups a day no matter how you split 'em up, fifty chin-ups in as few sets as possible, as many chin-ups as you can do in twenty minutes, slow chin-ups, explosive chest-to-bar chin-ups, chin-ups with a little kip on the last rep or two, chin-ups with a friend helping for a few extra reps, chin-up contests, chin-ups alone at home, chin-ups on alternating days, chin-ups at the beginning of the workout, chin-ups throughout the workout, chin-ups at the end of the workout, chin-ups for biceps, chin-ups for justice, chin-ups for God and country, chin-ups for looking good, chin-ups for athletics, chin-ups just for you, chin-up gumbo, chin-up lasagna, chin-up ice cream.
In conclusion, chin-ups. (And pull-ups.)
Chin-ups and pull-ups. If you don't have a pull-up bar, buy one, or go outside, parks and playgrounds will often have such facilities.
If you don't have a pull-up bar, or while you're waiting until it arrives, you can do isometric bicep exercises, using eg. a table (putting your hands under and pulling it up) or a towel (putting it eg. under your toes and pulling up).
If you mean bicep exercises with absolutely no equipment, just your body, then there's not much. You can still do isometric exercises, eg. pulling your legs, or knee to your chest (standing on one leg or lying on the floor) while simultaneously pushing your knees away. Another thing that works the bicep is turning your forearms while keeping the elbows fixed and bent. Note, that the biceps are stabilizers in most exercises where your body is supported on your hands (push-ups, handstands etc.) that is, they don't work much. But none of this compares in intensity to pull-ups (the last couple of exercises I mentioned are mostly used in rehab).