We study the approximability and inapproximability of finding identifying codes and locating-dominating codes of the minimum size. In general graphs, we show that it is possible to approximate both problems within a logarithmic factor, but sublogarithmic approximation ratios are intractable. In bounded-degree graphs, there is a trivial constant-factor approximation algorithm, but arbitrarily low approximation ratios remain intractable. In so-called local graphs, there is a polynomial-time approximation scheme. We also consider fractional packing of codes and a related problem of finding minimum-weight codes.