We prove the existence of global-in-time weak solutions to reaction-cross-diffusion systems for an arbitrary number of competing population species. These equations can be derived from an on-lattice random-walk model with general transition rates. In the case of linear transition rates, the model extends the two-species population model of Shigesada, Kawasaki, and Teramoto. The equations are considered in a bounded domain with homogeneous Neumann boundary conditions. Our existence result is based on a refined entropy method and a new approximation scheme. Global existence follows under a detailed balance or weak cross-diffusion condition, where detailed balance is related to the symmetry of the mobility matrix, which mirrors Onsager’s principle in thermodynamics. We can show that under detailed balance (and without reaction), the entropy is nonincreasing in time, but counter-examples suggest that the entropy may increase initially if detailed balance does not hold. This is a joint work with X. Chen and A. Juengel.