The study of algebras of bounded noncommutative analytic functions on subvarieties of noncommutative unit balls has led us to associate a spectral radius function with every finite-dimensional operator space E. Concretely, given a d-tuple of operators, the spectral radius is defined via a certain tensor power limit formula, which reduces to Gelfand’s spectral radius formula when E is one-dimensional. When E is the row operator space, for example, then our spectral radius coincides with the joint spectral radius studied by Bunce, Popescu and others. For every operator space E, the value of the spectral radius associated to E on given a tuple is strictly smaller than 1 if and only if the tuple is jointly similar to a tuple that lies in the open noncommutative unit ball corresponding to E. In this talk, based on joint work with Eli Shamovich, I will explain why we were led to this notion, present some examples, and describe possible applications. Finally, I will discuss recent progress on the case of commuting tuples, where several alternative notions of joint spectrum coincide, and pose some open problems in the noncommutative case.