Kronecker proved that every nonconstant polynomial has a root in a finite extension of the coefficient field. Analogous attempts to solve equations in the category of groups run into immediate obstacles that are intrinsic to group theory. (For example, elements of distinct finite orders can not be conjugate in any group.) Nevertheless, there are quite general positive results dating back more than half a century and the prospect for more is quite good. One approach has been to build a topological model an equation or system of equations and then translate the existence of solutions into a family of mapping problems. These and related mapping problems carry information that has significant impact for the structure of groups given by generators and relations.