In this chapter, we encounter the elegant and powerful concept of conformal measures, which is due to Patterson for Fuchsian groups and due to Sullivan for all Kleinian groups and rational functions of the Riemann sphere. We deal, in this chapter, with conformal measures in the settings of the previous chapter. Sullivan conformal measures and their invariant versions will form the central theme of Volume 2. In fact, the current chapter is the first and essential step for construction of Sullivan conformal measures for elliptic functions. It deals with holomorphic maps defined on some open neighborhood of a compact invariant subset of a parabolic Riemann surface. We provide a fairly complete account of Sullivan conformal measures in such a setting. We also introduce several dynamically significant concepts and sets such as radial or conical points and several fractal dimensions defined in dynamical terms. We relate them to exponents of conformal measures. However, choosing the most natural, at least in our opinion, framework, we do not restrict ourselves to conformal dynamical systems only but present, in the first section of this chapter, a fairly complete account of the theory of general conformal measures.