Affine processes are distinguished by their rich structural properties, which makes them favorite when it comes to computations in financial applications of all kind. This fact has been explored and illustrated for the time-homogeneous case in a recent paper by Duffie, Filipovic and Schachermayer. However, there are many situations which require time-dependent parameters, such as when it comes to model calibration. This paper provides a rigorous treatment and complete characterization of time-inhomogeneous affine processes