  We review some probabilistic properties of the sum-of-digits function of
random integers. New asymptotic approximations to the total variation distance
and its refinements are also derived. Four different approaches are used: a
classical probability approach, Stein's method, an analytic approach and a new
approach based on Krawtchouk polynomials and the Parseval identity. We also
extend the study to a simple, general numeration system for which similar
approximation theorems are derived.
