Tutte's embedding is one of the most popular approaches for computing parameterizations of surface meshes in computer graphics and geometry processing. Its popularity can be attributed to its simplicity, the guaranteed bijectivity of the embedding, and its relation to continuous harmonic mappings. In this work we extend Tutte's embedding into hyperbolic conesurfaces called orbifolds. Hyperbolic orbifolds are simple surfaces exhibiting different topologies and cone singularities and therefore provide a flexible and useful family of target domains. The hyperbolic Orbifold Tutte embedding is defined as a critical point of a Dirichlet energy with special boundary constraints and is proved to be bijective, while also satisfying a set of points-constraints. An efficient algorithm for computing these embeddings is developed. We demonstrate a powerful application of the hyperbolic Tutte embedding for computing a consistent set of bijective, seamless maps between all pairs in a collection of shapes, interpolating a set of user-prescribed landmarks, in a fast and robust manner.