In this chapter we describe Godunov-type explicit shock-capturing methods designed to compute fluid flows governed by the one-and two-dimensional shallow water equations. These methods are based on the solution of the corresponding one-dimensional Riemann problem. We present exact and approximate solvers for the shallow water equations. Among them are exact Riemann problem solver, and the approximate Courant-Isaacson-Rees, Roe, and Osher-Solomon solvers. All these solvers permit one to perform calculations involving an arbitrary bottom relief. Special attention is paid to the exact Riemann problem solver for the shallow water equations. It allows one to calculate flows with shoaling and dry bottom regions by the explicit shock-capturing method without applying any data-fitting techniques.