Summary: This paper is devoted to a Shannon-theoretic study of turbo codes. We prove that ensembles of parallel and serial turbo codes are good in the following sense. For a turbo code ensemble defined by a fixed set of component codes (subject only to mild necessary restrictions), there exists a positive number \(\gamma_0\) such that for any binary-input memoryless channel whose Bhattacharyya noise parameter is less than \(\gamma_0\), the average maximum-likelihood (ML) decoder block error probability approaches zero, at least as fast as \(n^{-\beta}\), where \(\beta\) is the ``interleaver gain'' exponent defined by Benedetto et al. in 1996.