  Modular and quasimodular forms have played an important role in gravity and
string theory. Eisenstein series have appeared systematically in the
determination of spectrums and partition functions, in the description of
non-perturbative effects, in higher-order corrections of scalar-field spaces,
... The latter often appear as gravitational instantons i.e. as special
solutions of Einstein's equations. In the present lecture notes we present a
class of such solutions in four dimensions, obtained by requiring (conformal)
self-duality and Bianchi IX homogeneity. In this case, a vast range of
configurations exist, which exhibit interesting modular properties. Examples of
other Einstein spaces, without Bianchi IX symmetry, but with similar features
are also given. Finally we discuss the emergence and the role of Eisenstein
series in the framework of field and string theory perturbative expansions, and
motivate the need for unravelling novel modular structures.
