How much information about a group G is contained in the group ring K(G) for an arbitrary field K? Can one recover the algebraic or geometric structure of G from the ring? Are the algorithmic properties of K(G) similar to that of G? I will discuss all these questions in conjunction with the classical Kaplansky-type problems for some interesting classes of groups, in particular, for limit, hyperbolic, and solvable groups. At the end I will touch on the solution to the generalized 10th Hilbert problem in group rings and how equations in groups are related to equations in the group rings. The talk is based on joint results with O. Kharlampovich.