We present a Kalman-style realization theory for linear parameter-varying state-space representations whose matrices depend on the scheduling variables in an affine way (abbreviated as LPV-SSA). We show that minimality of LPV-SSAs is equivalent to observability and span-reachability rank conditions, and that minimal LPV-SSAs of the same input-output map are isomorphic. We present necessary and sufficient conditions for existence of an LPV-SSA in terms of the rank of a Hankel-matrix and a Ho-Kalman-like realization algorithm.