  This expository article introduces the Kapustin-Witten equations to
mathematicians. We discuss the connections between the Complex Yang-Mills
equations and the Kapustin-Witten equations. In addition, we show the relation
between the Kapustin-Witten equations, the moment map condition and the
gradient Chern-Simons flow. The new results in the paper correspond to
estimates on the solutions to the KW equations given an estimate on the complex
part of the connection. This leaves open the problem of obtaining global
estimates on the complex part of the connection.
