A cluster algebra is a ring defined by combinatorial data: a quiver viewed up to mutation equivalence. The mutation graph of a (mutation) equivalence class of quivers has a vertex for each quiver and an edge between two vertices if their quivers are related by a mutation. A mutation cycle is a cycle in this graph. We show that already for 4-vertex quivers there are examples of mutation cycles that have arbitrarily large length and cannot be paved by shorter cycles, and discuss other constructions of mutation cycles. This talk is based on joint work with Sergey Fomin.