According to Hilbert, the theory of complex multiplication, which brings together number theory and analysis, is not only the most beautiful part of mathematics but also of all science. "Complex multiplication" refers to a lattice in the complex numbers (or an elliptic curve) which admits endomorphisms by a ring larger than the integers. We will begin with Kronecker's "Jugendtraum" -- the use of complex multiplication to solve Hilbert's twelfth problem. This will lead us into a discussion of Heegner points. We will conclude with some recent work on the modular curve "at infinite level", and the unexpected role that complex multiplication plays in its geometry.