Stein's method is used to prove approximations in total variation to the
distributions of integer valued random variables by (possibly signed)
compound Poisson measures. For sums of independent random variables,
the results obtained are very explicit, and improve upon earlier
work of Kruopis (1983) and Čekanavičius (1997);
coupling methods are used to derive concrete expressions for the error
bounds. An example is given to illustrate the potential for application
to sums of dependent random variables.