A robotic cella manufacturing system widely used in industrycontains two or more robot-served machines, repetitively producing a number of part types. In this paper, we consider scheduling of operations in a bufferless dual-gripper robotic cell processing multiple part types. The processing constraints specify the cell to be a flowshop. The objective is to determine the robot move sequence and the sequence in which parts are to be processed so as to maximize the long-run average throughput rate for repetitive production of parts. We provide a framework to study the problem, and address the issues of problem complexity and solvability. Focusing on a particular class of robot move sequences, we identify all potentially optimal robot move sequences for the part-sequencing problem in a two-machine dual-gripper robot cell. In the case when the gripper switching time is sufficiently small, we specify the best robot move sequence in the class. We prove the problem of finding an optimal part sequence to be strongly NP-hard, even when the robot move sequence is specified. We provide a heuristic approach to solve the general two-machine problem and evaluate its performance on the set of randomly generated problem instances. We perform computations to estimate the productivity gain of using a dual-gripper robot in place of a single-gripper robot. Finally, we extend our results for the two-machine cell to solve an m-machine problem.