  The global-in-time existence of renormalized solutions to
reaction-cross-diffu-sion systems for an arbitrary number of variables in
bounded domains with no-flux boundary conditions is proved. The cross-diffusion
part describes the segregation of population species and is a generalization of
the Shigesada-Kawasaki-Teramoto model. The diffusion matrix is not diagonal and
generally neither symmetric nor positive semi-definite, but the system
possesses a formal gradient-flow or entropy structure. The reaction part
includes reversible reactions of mass-action kinetics and does not obey any
growth condition. The existence result generalizes both the condition on the
reaction part required in the boundedness-by-entropy method and the proof of J.
Fischer for reaction-diffusion systems with diagonal diffusion matrices.
