A joint channel estimation and channel coding scheme is presented for channels with memory using polar codes. Unlike the conventional approach of first estimating all channel parameters and then performing channel decoding separately, the proposed scheme incorporates a subset of reliable estimates of channel parameters into the decoding procedure and computes decoding metrics averaged over the statistical behavior of the channel. Specifically, decoding algorithms for finite-state Markov channels of any order, for the Gauss-Markov channel and for flat-fading channels are presented. Further, by adapting list decoding to identify reliably-decoded bits within a codeword, channel estimation and decoding steps are performed iteratively to boost the reliability of channel estimation as well as error correction. In order to improve the performance even further, a new pilot arrangement scheme is developed that utilizes the structure of polar codes and sends pilot symbols embedded within the polar codewords. This construction can be viewed as a new family of shortened polar codes that can be of independent interest. Simulation results demonstrate the benefit of the proposed approach compared to existing solutions.