Period integrals, or integrals of Laplace eigenfunctions over submanifolds, give us just one of many ways of exploring the relationship between geometry and the asymptotic distribution of eigenfunctions over manifolds. The problem has its roots in the spectral theory of automorphic forms, where we are interested in the Fourier coefficients of eigenfunctions restricted to closed geodesics/circles/horocycles on hyperbolic surfaces. We discuss a brief history of related results including bounds of period integrals over curves, recent generalizations to higher dimensions and to Fourier coefficients of eigenfunctions along curves, and some conjectures and hard problems.