Linear parameterized models of optical flow, particularly affine models, have become widespread in image motion analysis. The linear model coefficients are straightforward to estimate, and they provide reliable estimates of the optical flow of smooth surfaces. Here we explore the use of parameterized motion models that represent much more varied and complex motions. Our goals are threefold: to construct linear bases for complex motion phenomena; to estimate the coefficients of these linear models; and to recognize or classify image motions from the estimated coefficients. We consider two broad classes of motions: i) generic “motion features” such as motion discontinuities and moving bars; and ii) non-rigid, object-specific, motions such as the motion of human mouths. For motion features we construct a basis of steerable flow fields that approximate the motion features. For object-specific motions we construct basis flow fields from example motions using principal component analysis. In both cases, the model coefficients can be estimated directly from spatiotemporal image derivatives with a robust, multi-resolution scheme. Finally, we show how these model coefficients can be use to detect and recognize specific motions such as occlusion boundaries and facial expressions.