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We describe coding techniques that achieve the capacity of a discrete memoryless asymmetric channel. To do so, we discuss how recent advances in coding for symmetric channels yield more efficient solutions also for the asymmetric case. In more detail, we consider three basic approaches. The first one is Gallager's scheme that concatenates a linear code with a non-linear mapper, in order to bias the input distribution. We explicitly show that both polar codes and spatially coupled codes can be employed in this scenario. Further, we derive a scaling law between the gap to capacity, the cardinality of channel input and output alphabets, and the required size of the mapper. The second one is an integrated approach in which the coding scheme is used both for source coding, in order to create codewords with the capacity-achieving distribution, and for channel coding, in order to provide error protection. Such a technique has been recently introduced by Honda and Yamamoto in the context of polar codes, and we show how to apply it also to the design of sparse graph codes. The third approach is based on an idea due to Bo&#x0308;cherer and Mathar and separates completely the two tasks of source coding and channel coding by &#x201C;chaining&#x201D; together several codewords. We prove that we can combine any suitable source code with any suitable channel code in order to provide optimal schemes for asymmetric channels. In particular, polar codes and spatially coupled codes fulfill the required conditions.

How to achieve the capacity of asymmetric channels