It is known that a matrix-valued transfer function P has a stabilizing dynamic controller Q ( i.e., [(I)(-Q)(-P)(I)](-1) is an element of H(infinity)) iff P has a right (or left) coprime factorization. We show that the same result is true in the operator-valued case. Thus, the standard Youla-Bongiorno parameterization applies to every dynamically stabilizable function. We then derive further equivalent conditions, one of them being that P has a stabilizing controller with internal loop; this and some others are new even in the scalar-valued case. We also establish certain related results. For example, we extend the classical results on coprime factorization and partial feedback (measurement-feedback) stabilization to nonrational transfer functions. All our results apply in both discrete- and continuous-time settings, except that in the latter it is not clear whether the controller Q can always be chosen so that it is "continuous-time proper" (holomorphic and bounded on a right half-plane) unless, e.g., P(z) -> 0 as Re z -> +infinity.