In this article we discuss Markovian term structure models in discrete time and with continuous state space. More precisely we are concerned with the structural properties of such models if one has the Markov property for a part of the forward curve. We investigate the two cases where these parts are either a true subset of the forward curve, including the short rate, or the entire forward curve. For the former case we give a sufficient condition for the term structure model to be affine. For the latter case we provide a version of the HJM [6] drift condition (see also [7]). Under a Gaussian assumption an HJM-Musiela [10] type equation is derived