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The homeomorphism problem for closed 3-manifolds was essentially solved after Perelman's work proving the Geometrization Conjecture. The work of many previous authors is put together and different algorithms are used to deal with various cases. In this talk, I will start by explaining the general picture. In the case of two closed hyperbolic 3-manifolds, the homeomorphism problem was solved by appealing to Sela's solution of the isomorphism problem for torsion free word hyperbolic groups. In joint work with Hamish Short, we have given a more geometric algorithm for this case which avoids appealing to Sela's algebraic result. Our work is an extension of earlier work of Manning.