Analytical, three-dimensional Wannier functions have been constructed from Bloch states formed from a lattice of gaussians. By considering a natural extension of the three-dimensional Brillouin zone to a six-dimensional complex space, the authors are able to investigate the localisation of the functions. The localisation and asymptotic behaviour are determined by examining the zeros of the Fourier transform of the atomic overlap matrix. They discuss in detail two systems, based on the simple-cubic and body-centred-cubic lattices respectively. They find both cases exhibit faster than exponential localisation. It is shown that the localisation of the three-dimensional functions depends on the angle of the asymptotic direction relative to the host lattice as well as on an exponential of the radial distance. A simple interpretation is developed: three-dimensional Wannier functions are least localised in the direction of the nearest perturbing neighbour atoms. Wannier functions from a one-dimensional lattice of gaussians have been introduced earlier by Wannier. The authors show analytically they decay asymptotically like mod x mod ^-1/2 exp(- pi phi mod x mod /4), where phi is the gaussian localisation parameter, and also that they oscillate in phase with the host lattice but are phase-shifted away from the atomic positions except for the zeroth site. They show that the behaviour in three dimensions is similar.