This paper attempts to provide a comprehensive, coherent, and methodical review of the existing literature on the transient performance of extended surfaces (fins). With the exception of very few experimental studies, the work described is theoretical involving either analytical approaches or numerical computations. For both longitudinal and radial convecting fins, the shapes analyzed include fins of rectangular, trapezoidal, concave parabolic, and convex parabolic profiles. The boundary conditions considered are those of time-dependent base temperature or time-dependent base heat flow, or time-dependent temperature of the fluid on the unfinned side of the primary surface. Other studies discussed consider the transient responses of radiating and convecting-radiating fins, fins with stochastic base temperature and fins in which conduction, natural convection, and radiation are coupled. The paper also provides an in-depth consideration of two-dimensional analyses of a cylindrical spine (pin fin) and a composite longitudinal fin. All of the important results pertaining to the transient effect in extended surfaces are presented and these effects are illustrated with several examples. The paper concludes with an attempt to identify future investigative efforts.