Circle patterns are one of the ways to uniformize graphs on surfaces, by embedding them in such a way that every face admits a circumcircle. In this talk I will describe a discrete-time dynamical system on circle patterns with the combinatorics of the square grid, called Miquel dynamics. It is based on the classical Miquel's six circles theorem. I will present some properties of this dynamics which suggest its integrability. \\ \\ Partly joint work with Alexey Glutsyuk (CNRS, Ecole normale superieure de Lyon / Higher School of Economics).