This paper considers scheduling problems arising in robot-served manufacturing cells in which the machines are configured in a flowshop that repetitively produces a family of similar parts. We study the problem of determining the robot move cycle and the part sequence that jointly minimize the average steady-state cycle time required for the repetitive production of a minimal part set, or equivalently maximize the long-run throughput rate. Three earlier related papers provide algorithms, or proofs of intractability, for a variety of cell configurations. We use the intuition developed there to design and test simple heuristic procedures for the part sequencing problem under different robot move cycles in three-machine cells where that problem is intractable. This enables us to develop a heuristic procedure for a general three-machine cell. A methodology for extending this heuristic to four-machine cells is described and tested. Ideas for extension to even larger cells are also discussed. We also describe and test two heuristics for a cell design problem that involves partitioning machines into cells as well as determining the sequence of robot moves and parts. Finally, we provide a list of open research problems.