Research on organic thin-film transistors tends to focus on improvements in device performance, but very little is understood about the ultimate limits of these devices, the microscopic physical mechanisms responsible for their limitations, and, more generally, the intrinsic transport properties of organic semiconductors. These topics are now being investigated through the study of transport in organic transistors realized using molecular single crystals of unprecedented chemical purity and structural quality. These studies are elucidating detailed microscopic aspects of the physics of organic semiconductors and corresponding devices and have also led to unforeseen high values for carrier mobility in these materials. Here, we discuss developments in this area and present a brief outlook on future goals that have come into experimental reach.