  We give a survey of the long-time asymptotics for the Toda lattice with
steplike constant initial data using the nonlinear steepest descent analysis
and its extension based on a suitably chosen $g$-function. Analytic formulas
for the leading term of the asymptotic solutions of the Toda shock and
rarefaction problems (including the case of overlapping background spectra) are
given and complemented by numerical simulations. We provide an explicit formula
for the modulated solution in terms of Abelian integrals on the underlying
hyperelliptic Riemann surface.
