The second part of Problem 16 in Hilbert's famous list deals with limit cycles of polynomial differential equations on $\mathbb{R}^2$. Limit cycles were first studied by Poincare in his 1881 work Memoire sur les courbes definies par une equation differentielle and lie in the heart of the qualitative theory of planar systems. This apparently simple but remarkably challenging problem has had a dramatic and exciting history. In this talk I intend to give a brief survey on the history of the problem and the influence it has had on the qualitative theory of dynamical systems.