Mismatched decoding models the coded communication problem in scenarios where a sub-optimal decoder is used. These situations arise when optimal maximum-likelihood decoding cannot be used: i) the channel transition is unknown and imperfectly estimated or, ii) when, for complexity reasons, the channel likelihood is too difficult to compute and an alternative decoding metric is needed. In addition, important problems in information theory such as zero-error capacity or erasures-only capacity can be cast as instances of mismatched decoding. In the mismatched decoding problem, the optimal maximum-likelihood decoder is replaced by a maximum metric decoder, where the metric is not necessarily the channel likelihood. This thesis studies the problem of channel coding with mismatched decoding. Single-letter characterization of mismatch capacity is a long standing open problem in information theory. Lower bounds for the mismatch capacity have been studied extensively using random coding techniques. On the other hand, upperbounds for the mismatch capacity were not explored until a few years ago. In this thesis we study a novel technique for tackling this problem based on analysing the probability of error of a given codebook using an auxiliary channel and a hypothetical decoder. Careful design of the auxiliary channel leads to a lower bound on the probability of error of the codebook in the original mismatched decoding problem. The first version of our upper bound based on the mentioned technique studies the case where we design the auxiliary channel with the following property. When a codeword is sent over the auxiliary channel and the hypothetical decoder makes an error then the mismatched decoder makes an error for sure. The results is this part are mainly derived using the method of types and graph theory. Moreover, the resulting bound is shown to be computable using convex optimization techniques. A simple algorithm is introduced and is shown to converge to the derived bound. A multiletter version of the bound is studied and is shown to yield no improvement over the single-letter bound. An implication of the bound for binary-input channels is derived which provides a simple sufficient condition for the mismatch capacity being strictly less than the matched capacity. Finally, using some examples the application of the bound is illustrated. A generalized version of the mentioned method is used to derive an upper bound to the reliability function. In contrast to the previous part, when a codeword is sent over the auxiliary channel and the hypothetical decoder makes an error, the mismatched decoder makes an error with a positive probability. Moreover, this probability is conditioned to be bounded away from zero by the inverse of a polynomial function of the block length. This is a strictly weaker design assumption for the auxiliary channel and the hypothetical decoder compared to the previous one. Nevertheless, the rate that the reliability functions becomes equal to zero is shown to be an upper bound to the mismatch capacity. To prove the validity of this upper bound on the reliability function the method of types and graph theory are mainly used. The optimization problem for computing this upper bound is shown to be non-convex. This upper bound is compared to the existing upper bounds on the mismatch capacity and is shown to improve over the recent ones. Using several numerical examples the application of the bound is illustrated.