The theory of complex multiplication occupies an important place in number theory, an early manifestation of which was the use of special values of the j-function in explicit class field theory of imaginary quadratic fields, and the works of Eisenstein, Kronecker, Weber, Hilbert, and many others. In the early 20th century, Hecke studied the diagonal restrictions of Eisenstein series over real quadratic fields, which later lead to highly influential developments in the theory of complex multiplication initiated by Gross and Zagier in their famous work on Heegner points on elliptic curves. In this talk, we will explore what happens when we replace the imaginary quadratic fields in CM theory with real quadratic fields, and propose a framework for a tentative 'RM theory', based on the notion of rigid meromorphic cocycles, introduced in joint work with Henri Darmon. I will discuss some recent progress obtained in various joint works with Henri Darmon, Alice Pozzi, and Yingkun Li.