Error-correcting capabilities of concatenated codes with maximum distance separable (MDS) outer codes and time-varying inner codes, used on memoryless discrete channels with maximum-likelihood decoding, are investigated. It is proved that, asymptotically, the Gallager random coding theorem can be obtained for all rates by such codes. Further, the expurgated coding theorem, as well, is proved to be valid for all rates on regular channels. The latter result implies that the Gilbert-Varshamov bound for block codes over any finite field can be obtained asymptotically for all rates by linear concatenated codes.