Summary: Waves in shallow water are computed by moving blocks of water in the direction of the flow using a Lagrangian method. The mass and momentum in the displaced-and-deformed blocks after the Lagrangian advection are re-distributed back on to the Eulerian mesh to form new blocks at every increment of time. This Lagrangian block advection guarantees for positive water depth. It also prevents the occurrence of unphysical numerical oscillations. Several numerically challenging problems are considered in a series of simulations using the method. The first problem is the tracking of wetting-and-drying interface in a parabolic bowl. The second problem is the capture of depth and velocity discontinuities across the shock waves. Finally, the block advection method is applied to calculate the flood waves overtopping a meandering river. The results of the simulations are compared with the exact solutions. The convergence of Lagrangian block advection towards the exact solutions is first-order accurate in the simulations of the depth-and-velocity discontinuities.