The rotary-vee engine is a novel and unusual internal combustion engine. The rotary-vee engine is unique in that all of the components have rotary motion, but the combustion chamber and piston design is similar to a reciprocating engine. Of particular significance, the rotary-vee engine design includes pistons with rings to accomplish the sealing of the combustion chamber. Thus, the rotary-vee engine may offer the sealing benefits of the conventional piston engine, and the vibration and balance characteristics of a rotary engine. This paper includes a review of rotary engines, and places the rotary-vee engine in the context of all rotary engines. In addition, a thermodynamic analysis of the operation of a rotary-vee engine is reported. The rotary-vee engine possesses some advantages relative to other rotary engine designs such as piston ring sealing, and the thermodynamic analysis indicates similar performance as compared to conventional reciprocating engines.