Of all the conceivable ideal-gas processes, almost all introductory physics textbooks as well as more advanced texts on thermodynamics emphasize only four: isochoric, isobaric, isothermal, and adiabatic (isentropic). These are processes in which a state variable--volume, pressure, temperature, or entropy--remains constant. It turns out that these four processes are examples of a more general ideal-gas process, called a "polytropic process," in which the specific heat remains constant. Polytropic processes are frequently discussed in engineering thermodynamics but rarely in physics. The goal of this paper is to bring polytropic processes to the attention of physics instructors. These processes have interesting properties (such as, in some cases, negative specific heat), they are easily accessible at the introductory physics level, and they expand the range of examples and problems that can be explored in thermodynamics.