In this article, we define operator algebras internal to a rigid C*-tensor category $\mathcal{ C}$. A C*/W*-algebra object in $\mathcal{ C}$ is an algebra object $\mathbf{ A}$ in $\operatorname{ ind}$-$\mathcal{ C}$ whose category of free modules ${ \sf FreeMod}_{ \mathcal{ C}}(\mathbf{ A})$ is a $\mathcal{ C}$-module C*/W*-category respectively. When $\mathcal{ C}={ \sf Hilb_{ f.d.}}$, the category of finite dimensional Hilbert spaces, we recover the usual notions of operator algebras. We generalize basic representation theoretic results, such as the Gelfand-Naimark and von Neumann bicommutant theorems, along with the GNS construction. We define the notion of completely positive maps between C*-algebra objects in $\mathcal{ C}$ and prove the analog of the Stinespring dilation theorem. As an application, we discuss approximation and rigidity properties, including amenability, the Haagerup property, and property (T) for a connected W*-algebra $\mathbf{ M}$ in $\mathcal{ C}$. Our definitions simultaneously unify the definitions of analytic properties for discrete quantum groups and rigid C*-tensor categories.
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