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Many convex optimization problems can be represented through conic extended formulations with auxiliary variables and constraints using only the small number of standard cones recognized by advanced conic solvers such as MOSEK 9. Such extended formulations are often significantly larger and more complex than equivalent conic natural formulations, which can use a much broader class of exotic cones. We define an exotic cone as a proper cone for which we can implement tractable logarithmically homogeneous self-concordant barrier oracles for either the cone or its dual cone. In this paper we introduce Hypatia, a highly-configurable open-source conic primal-dual interior point solver with a generic interface for exotic cones. Hypatia is written in Julia and accessible through JuMP, and currently implements around two dozen useful predefined cones (some with multiple variants). We define some of Hypatia's exotic cones, and for conic constraints over these cones, we analyze techniques for constructing equivalent representations using the standard cones. For optimization problems from a variety of applications, we introduce natural formulations using these exotic cones, and we show that the natural formulations are simpler and lower-dimensional than the equivalent extended formulations. Our computational experiments demonstrate the potential advantages, especially in terms of solve time and memory usage, of solving the natural formulations with Hypatia compared to solving the extended formulations with either Hypatia or MOSEK 9.