The origins of Carleman estimates lie with the pioneering 1939 work by the Swedish mathematician T. Carleman, concerned with the unique continuation property for solutions for linear elliptic partial differential equations with smooth coefficients in dimension two. The fundamental new idea introduced by Carleman, which consists of establishing a priori energy estimates involving an exponential weight, has permeated essentially all the subsequent work in the subject. More recently, Carleman estimates have found numerous striking applications beyond the original domain of unique continuation, from control theory to spectral theory to the analysis of inverse problems. The purpose of this talk is to provide a broad introduction to the subject and to attempt to illustrate some of its inner workings.