The notion of an algebra makes sense in any monoidal category C4. When the monoidal category has extra structure, we can endow our algebra with the corresponding structure. We will be interested in defining the notion of a C*4 or W*4 (von Neumann) algebra internal to a category C. We will begin by defining a (concrete) rigid C*- tensor category,and highlight the important features for the abstract setting. We will then define a C*/W*- algebra in C in terms of modules for that algebra. One can prove many analogs of basic representation theoretic results for ordinary operator algebras, like the Gelfand-Naimark Theorem or the Stinespring Dilation Theorem. This talk is based on joint work with Corey Jones (arXiv:1611.04620).