The spans of ann-step random walk on a simple cubic lattice are the sides of the smallest rectangular box, with sides parallel to the coordinate axes, that contains the random walk. Daniels first developed the theory in outline and derived results for the simple random walk on a line. We show that the development of a more general asymptotic theory is facilitated by introducing the spectral representation of step probabilities. This allows us to consider the probability density for spans of random walks in which all moments of single steps may be infinite. The theory can also be extended to continuous-time random walks. We also show that the use of Abelian summation simplifies calculation of the moments.