For some hybridizable discontinuous Galerkin (HDG) methods, suitably devised projections make their analyses simple and concise. However, devising these projections is usually difficult and many important HDG methods still lack their corresponding projections; consequently, their analyses become cumbersome. In this thesis, we propose novel analytical tools to solve this problem. These tools can be used to systematically devise and analyze new HDG methods, to unify their analyses, and to simplify and improve existing ones. We shall study these tools and their applications in three cases: (1) HDG methods for elastic problems, (2) HDG methods on polyhedral meshes, and (3) HDG methods for Maxwell equations. They will be discussed in Chapter 2, Chapter 3, and Chapter 4, respectively. In Chapter 1, we give an introduction to motivate the topic of this thesis. Finally in Chapter 5, we conclude by discussing several promising potential developments of our work.