The field of microelectromechanical systems (MEMS) for photonic integrated circuits (PICs) is reviewed. This field leverages mechanics at the nanometer to micrometer scale to improve existing components and introduce novel functionalities in PICs. This review covers the MEMS actuation principles and the mechanical tuning mechanisms for integrated photonics. The state of the art of MEMS tunable components in PICs is quantitatively reviewed and critically assessed with respect to suitability for large-scale integration in existing PIC technology platforms. MEMS provide a powerful approach to overcome current limitations in PIC technologies and to enable a new design dimension with a wide range of applications.