The problem of low-density parity-check (LDPC) decoding in channels with memory has been attracting increasing attention in the literature. In this thesis, we use LDPC codes in an estimation-decoding scheme for the Gilbert-Elliott (GE) channel and more general Markov-modulated channels. The major accomplishments of this thesis include both analysis and design components. To analyze our estimation-decoding scheme, we derive density evolution for the GE channel, which has previously been used largely in memoryless channels. Furthermore, we develop techniques that use density evolution results to more efficiently characterize the space of Markov-modulated parameters. We begin by applying a characterization to the GE channel, following which we generalize this characterization into a partial ordering of Markov-modulated channels in terms of probability of symbol error. We also consider the design problem of developing LDPC degree sequences that are optimized for the GE channel. We obtain a novel design tool that approximates density evolution for our estimation-decoding algorithm, and present degree sequences that represent the best known codes in the GE channel. We also present a method of generalizing this tool to Markov-modulated channels, and give some of the first optimized degree sequences ever obtained for these channels.