In this talk I will discuss two tiling problems. The first problem is the problem of classifying all convex pentagons that admit tilings of the plane. This classification was recently determined, and I will present the history of the problem and some of my contributions to its solution. The second problem, called Heesch's Tiling Problem, concerns shapes that do not admit tilings of the plane, but copies of which can form a number of layers (coronas) of tiles around a centrally placed copy. Heesch's Tiling Problem, which remains unsolved, is connected to several central problems in the theory of tilings. I will present the history of this problem along with my contributions to it.