Manipulation of microscopic objects, especially biological objects and microelectromechanical systems (MEMS) components, has become an important area of robotics research over the past several years. Automation is necessary as it is challenging to manually control the microobjects due to the scaling effect of the surface forces, stochastic motion of objects in fluid media, and uncertainty associated with object state estimation. Automation requires real-time control of the position, orientation, and force applied by each of the operational manipulators, as governed by the system-level objectives of optimizing resource, time, and effort, by planning suitable actions for the manipulated objects. In this paper, we provide a survey of the research in planning and control of such automated micromanipulation operations. We present a broad taxonomy based on the underlying approach, and discuss the salient features and experimental success of each research effort. We also identify the major limitations and common trends across all the approaches, discuss the effectiveness of an approach depending on the operation characteristics, and outline promising future research directions.