Abstract: Vladimir Voevodsky was a brilliant mathematician, a Fields Medal winner, and a faculty member at the Institute for Advanced Study, until his sudden and unexpected death in 2017 at the age of 51. He had a special flair for thinking creatively about ways to incorporate topology and homotopy theory into other fields of mathematics. In this talk for a general audience, I will sketch his seminal contributions to two broad areas, algebraic geometry and the foundations of mathematics. A colleague commented about his work in the former area, which deals with polynomial equations in various alternative number systems, that if mathematics were music, then Voevodsky would be a musician who invented his own key to play in. His work in the latter area has led to a new alternative foundation for all of mathematics, opening up a new landscape populated by fundamental objects unseen in the traditional foundation provided by set theory, and in which the notion of equality is interpreted in an unexpected way. It also hastens the day when our mathematical literature has been verified mechanically and referees are relieved of the tedium of checking the proofs in articles submitted for publication.