In this work, we formalize the essential aspects of a (channel) coding theory in order to generalize the subject area to cover a broader collection of mathematical settings. After reviewing the classical theory within this new framework, we cover generalizations in three different directions. We examine block codes over nonabelian groups, where we prove that there are no nonabelian groups whose subgroup block codes support a duality theory, solving an open problem of Dougherty, Kim, and Sol{\'e}. This result can be rephrased in terms of group theory to say that given a finite group $G$, $G^n$ has a self-dual subgroup lattice for all $n$ if and only if $G$ is abelian. We define three different permutation coding theories under the Hamming, Kendall tau, and Ulam metrics. In each case, we review what is known about the Hamming and Singleton bounds in these coding theories. For the Kendall tau and Ulam metrics, we provide new insights into the weight enumerators of various subgroups, and demonstrate that for non-normal subgroups the weight distributions can vary significantly under conjugation. We will then study the relabeling that arises from this conjugation action, giving formulas for the minimum and maximum Kendall tau and Ulam weights of a permutation with a given cycle type, and we will apply these theorems to understand the minimum distances of conjugates of subgroup codes under these metrics. We will also introduce a sequence of $W_n^k$ numbers giving the maximum number of inversions in a permutation on $n$ symbols with a single $k$-cycle and $n-k$ fixed points. Finally, we'll present an entirely new theory of coding curves inspired by recent research into time-of-flight cameras as well as a more intuitive question of packing ropes into boxes. We will produce several basic bounds in this setting including a Hamming-type bound, and we will provide examples of lattice coding curves that approach these bounds in low dimensions. We also present work on feasible regions, connections to sphere packing, and Manin-type curves. We will finish this section with a brief discussion of several different directions to extend this theory.