  The classical theory of elliptic curves with complex multiplication is a
fundamental tool for studying the arithmetic of abelian extensions of imaginary
quadratic fields. While no direct analogue is available for real quadratic
fields, a (conjectural) theory of "real multiplication" was recently proposed
by Darmon and Vonk, relying on $p$-adic methods, and in particular on the new
notion of rigid meromorphic cocycles. A rigid meromorphic cocycle is a class in
the first cohomology of the group $\mathrm{SL}_2(\mathbb{Z}[1/p])$ acting on
the non-zero rigid meromorphic functions on the Drinfeld $p$-adic upper half
plane by M\"obius transformation. The values of rigid meromorphic cocycles at
real quadratic points can be thought of as analogues of singular moduli for
real quadratic fields. In this survey article, we will discuss aspects of the
theory of complex multiplication and compare them with conjectural analogues
for real quadratic fields, with an emphasis on the role played by families of
modular forms in both settings.
