Moonshine', in Mathematics, refers to surprising and deep connections between finite group theory and the theory of so-called modular forms. The first known instance of Moonshine is Monstrous Moonshine, where the coefficients of the modular j-function are identified as dimensions of representations of the Monster group. This was first observed by John McKay in 1987; Richard Borcherds received a Fields Medal for his proof of the resulting Moonshine Conjectures in 1998. More than a decade later, Tohru Eguchi, Hiroshi Ooguri and Yuji Tachikawa proposed 'Mathieu Moonshine', which links the largest Mathieu group to topological invariants of K3 surfaces, yielding the Fourier coefficients of a certain elliptic modular form. Conformal field theory turns out to be key to every known instance of moonshine. The talk will give an introduction to solved and unsolved mysteries of these two types of Moonshine.