In this paper we discuss AWGN coding theorems for ensembles of coding systems which are built from fixed convolutional codes interconnected with random interleavers. We call these systems “turbo-like” codes and they include as special cases both the classical turbo codes [1, 2, 3] and the serial concatentation of interleaved convolutional codes [4]. We offer a general conjecture about the behavior of the ensemble (maximum-likelihood decoder) word error probability as the word length approches infinity. We prove this conjecture for a simple class of rate 1/q serially concatenated codes where the outer code is a q-fold repetition code and the inner code is a rate 1 convolutional code with transfer function 1/(1+ D). We believe this represents the first rigorous proof of a coding theorem for turbo-like codes.