Abstract: By 1693, Leibniz realized that it is much easier to decide whether two polynomials have a common root than it is to find a root of one. Since that day the theory of resultants, or ``elimination theory'', recast in different ways, has played an important part in algebra and algebraic geometry. Work of Cayley and Grothendieck explained the appearance of determinants, and alternating products of determinants, in resultant formulas. Recent discoveries have simplified the picture further, and allowed a new interplay with the modern theory of vector bundles and with syzygies over exterior algebras. I will introduce resultants and explain some of the recent developments.