The study of the structure of the level sets of a harmonic function in the plane up to a homeomorphism is a classical problem that goes back to Morse, Kaplan and Boothby in the first half of the last century. In higher dimensions, there is a striking lack of significant results in this direction, the reasons of which are both analytical (as harmonic functions are no longer related to holomorphic functions of one complex variable) and topological (since the possible topologies of level sets should become increasingly complex). Our motivation is to solve very visual questions such as whether there are harmonic functions in R3 having level sets of arbitrarily high genus or whether one can take (for example) seven harmonic functions in R14 whose zero level sets intersect transversally at a Milnor 7-sphere. In this talk I will review some recent results in this direction, obtained in collaboration with D. Peralta-Salas, which, in particular, answer these questions in the affirmative. Generalizations to other elliptic equations and applications will be discussed as well.