abstract: Mapping class groups and their arithmetic analogues are fundamental groups of moduli spaces of curves. They can be built up from mapping class groups of genus 0 and 1. So, to understand the motivic structure of all mapping class groups, it should suffice to understand them in genera 0 and 1. The motivic structure of genus 0 mapping class groups is reasonably well understood, thanks to the work of Goncharov, Deligne and Brown. Genus 1 mapping class groups are (basically) modular groups. In this talk I will discuss motivic structures (Hodge and Galois) on completions of modular groups and their arithmetic analogues. These structures are remarkably rich due to the role of modular forms. I will conclude with a description of the theory of universal mixed elliptic motives. Much of this work is joint with Makoto Matsumoto.