The study of the cone of excessive measures associated with a Markov process goes back to Hunt's fundamental mem- oir [H57]. However until quite recently it received much less attention than the cone of excessive functions. The fact that an excessive function can be composed with the underlying Markov process to give a supermartingale, subject to secondary finiteness hypotheses, is crucial in the study of excessive func- tions. The lack of an analogous construct for excessive mea- sures seemed to make them much less tractable to a proba- bilistic analysis. This point of view changed radically with the appearance of the pioneering paper by Fitzsimmons and Maisonneuve [FM86] who showed that a certain stationary process associated with an excessive measure could be used to study excessive measures probabilistically. These station- ary processes or measures had been constructed by Kuznetsov [Ku74] extending earlier work of Dynkin. It is now common to call them Kuznetsov measures. Following the Fitzsimmons- Maisonneuve paper there was renewed interest and remarkable progress in the study of excessive measures. The purpose of this monograph is to organize under one cover and prove under standard hypotheses many of these recent results in the theory of excessive measures. The two basic tools in this recent development are Kuznet- sov measures mentioned above and the energy functional.