Variation in music is defined as repetition of a theme, but with various modifications, playing an important role in many musical genres in developing core music ideas into longer passages. Existing research on variation in music is mostly confined to datasets consisting of classical theme-and-variation pieces, and generative models limited to melody-only representations. In this paper, to address the problem of the lack of datasets, we propose an algorithm to extract theme-and-variation pairs automatically, and use it to annotate two datasets called POP909-TVar (2,871 theme-and-variation pairs) and VGMIDI-TVar (7,830 theme-and-variation pairs). We propose both non-deep learning and deep learning based symbolic music variation generation models, and report the results of a listening study and feature-based evaluation for these models. One of our two newly proposed models, called Variation Transformer, outperforms all other models that listeners evaluated for "variation success", including non-deep learning and deep learning based approaches. An implication of this work for the wider field of music making is that we now have a model that can generate material with stronger and perceivably more successful relationships to some given prompt or theme.