This paper investigates manipulation tasks with arrays of microelectromechanical structures (MEMS). We develop a geometric model for the mechanics of microactuators and a theory of sensorless, parallel manipulation, and we describe efficient algorithms for their evaluation. The theory of limit surfaces offers a purely geometric characterization of microscale contacts between actuator and moving object, which can be used to efficiently predict the motion of the object on an actuator array. It is shown how simple actuator control strategies can be used to uniquely align a part up to symmetry without sensor feedback. This theory is applicable to a wide range of microactuator arrays. Our actuators are oscillating structures of single-crystal silicon fabricated in a IC-compatible process. Calculations show that these actuators are strong enough to levitate and move, for example, a piece of paper.>