The Yang-Baxter equation appear in various situations in physics and mathematics. For example it arises as a consistency condition in integrable models. The reflection equation (boundary Yang-Baxter equation) is a generalization of the Yang-Baxter equation to systems with a boundary. A further generalization to systems with defects which admits both reflection and transmission can be made, which results in reflection-transmission Yang-Baxter equations.In this thesis the Yang-Baxter equation and the reflection equation are presented. Representations of the Temperley-Lieb algebra and the blob algebra are used to construct matrices which solve the respective equations. For the reflection-transmission Yang-Baxter equations, steps toward a solution are taken by using a similar approach as for the first two cases, namely by finding an algebra whose representations can be used to construct matrices which solve the equations.