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Cross-diffusion systems describe the diffusive interaction in multicomponent systems. Examples include population dynamics, cell biology, and thermodynamics. Mathematically, they consist of strongly coupled parabolic equations with a full diffusion matrix. The major challenge is that the diffusion matrix is generally neither symmetric nor positive definite, such that even local-in-time existence of solutions is far from being trivial. Recently, entropy methods have been developed to analyze cross-diffusion systems. The key idea is to formulate the equations using entropy variables (or chemical potentials) and to exploit Lyapunov functional techniques. In this talk, we introduce the boundedness-by-entropy method, we show how it leads to global-in-time existence of weak solutions to cross-diffusion systems from biology and physics, and we present some open problems.