  We prove a colorful generalization of the Borsuk--Ulam theorem and derive
colorful consequences from it, such as a colorful generalization of the ham
sandwich theorem. Even in the uncolored case this specializes to a
strengthening of the ham sandwich theorem, which given an additional condition,
contains a result of B\'{a}r\'{a}ny, Hubard, and Jer\'{o}nimo on well-separated
measures as a special case. We prove a colorful generalization of Fan's
antipodal sphere covering theorem, we derive a short proof of Gale's colorful
KKM theorem, and we prove a colorful generalization of Brouwer's fixed point
theorem. Our results also provide an alternative between Radon-type
intersection results and KKM-type covering results. Finally, we prove colorful
Borsuk--Ulam theorems for higher symmetry.
