  In the present text we give a geometric interpretation of quasi-modular forms
using moduli of elliptic curves with marked elements in their de Rham
cohomologies. In this way differential equations of modular and quasi-modular
forms are interpreted as vector fields on such moduli spaces and they can be
calculated from the Gauss-Manin connection of the corresponding universal
family of elliptic curves. For the full modular group such a differential
equation is calculated and it turns out to be the Ramanujan differential
equation between Eisenstein series. We also explain the notion of period map
constructed from elliptic integrals. This turns out to be the bridge between
the algebraic notion of a quasi-modular form and the one as a holomorphic
function on the upper half plane. In this way we also get another
interpretation, essentially due to Halphen, of the Ramanujan differential
equation in terms of hypergeometric functions. The interpretation of
quasi-modular forms as sections of jet bundles and some related enumerative
problems are also presented.
