The goal of this article is to describe the main ideas behind the new class of codes called turbo codes, whose performance in terms of bit error probability has been shown to be very close to the Shannon limit. In this article, the mathematical measures of a posteriori probability and likelihood are reviewed, and the benefits of turbo codes are explained in this context. Since the algorithms needed to implement the decoders have been well documented by others, they are only referenced here, not described in detail. A numerical example, using a simple concatenated coding scheme, provides a vehicle for illustrating how error performance can be improved when soft outputs from the decoders are used in an iterative decoding process.