This review provides an in-depth examination of the current status of research in the burgeoning field of spin caloritronics, which examines the relationships between spin and heat transport in materials. Within the context of energy conversion applications, thermal spin transport provides conceptually new mechanisms for solid-state thermal-to-electrical energy conversion that may be used for waste heat recovery and temperature control. In order to develop an appropriate background for our discussion of these new developments, we begin with a brief introduction to both spin-independent and spin-dependent transport phenomena. From there, we describe the various processes through which thermal spin currents can be actively generated, manipulated, and detected, with an emphasis on the recent discoveries related to the spin Seebeck effect. A description of spin transport in magnetic and non-magnetic insulators, semiconductors, and metals is provided, along with a discussion of various spin injection methods and geometries. We conclude by providing several examples of proposed thermal energy conversion devices based on spin caloritronic effects, as well as a brief outline of ongoing research challenges and open questions in this nascent field.