The Fourier transform established duality between two important bases: plane waves (localized in momentum) and delta-functions (localized in space). While Bloch functions are periodic case analogs of plane waves, Wannier functions are the Floquet theory analogs of delta-functions. The Wannier functions cannot be compactly supported, but one tries to obtain them with fast (up to exponential) decay, since this leads to their extreme usefulness in computational physics. Alas, as established more than four decades ago by D. Thouless, a topological obstacle can arise (and does arise in the presence of magnetic potentials), prohibiting existence of such a basis. Namely, the so-called Bloch bundle must be (albeit not always is) trivial. The chapter contains description and results arising from two approaches. The first one tries to establish conditions under which such obstacles do not arise, while the second one tries to resolve this issue by finding in the presence of the obstacle excellent overdetermined sets of Wannier functions, which are as useful as orthonormal bases.