Combining the nonvolatile, locally switchable polarization field of a ferroelectric thin film with a nanoscale electronic material in a field effect transistor structure offers the opportunity to examine and control a rich variety of mesoscopic phenomena and interface coupling. It is also possible to introduce new phases and functionalities into these hybrid systems through rational design. This paper reviews two rapidly progressing branches in the field of ferroelectric transistors, which employ two distinct classes of nanoscale electronic materials as the conducting channel, the two-dimensional (2D) electron gas graphene and the strongly correlated transition metal oxide thin films. The topics covered include the basic device physics, novel phenomena emerging in the hybrid systems, critical mechanisms that control the magnitude and stability of the field effect modulation and the mobility of the channel material, potential device applications, and the performance limitations of these devices due to the complex interface interactions and challenges in achieving controlled materials properties. Possible future directions for this field are also outlined, including local ferroelectric gate control via nanoscale domain patterning and incorporating other emergent materials in this device concept, such as the simple binary ferroelectrics, layered 2D transition metal dichalcogenides, and the 4d and 5d heavy metal compounds with strong spin-orbit coupling.