There have been some major advances in the theory of optimal designs for interference models. However, the majority of them focus on one-dimensional layout of the block and the study for two-dimensional interference model is quite limited partly due to technical difficulties. In this thesis, we try to fill this gap by systematically characterizing all possible universally optimal designs simultaneously. Our research in such direction is presented in five chapters. Chapter 1 introduces the backgrounds and the motivations of our research. A review of the works in the literature is also included. Chapter 2 gives the optimal and efficient designs under the criteria of universally optimal. We first explain the interference model in details. We then introduce the complete class and drive a necessary and sufficient condition for a design within it to be universally optimal. Later, we establish a necessary and sufficient condition for an arbitrary design to be universally optimal. The extended case of different side effects is also addressed. Chapter 3 derive theoretical results regarding the supporting set of block arrays. This shrinks the pool of feasible designs and saves the computational cost tremendously. Chapter 4 provides some concrete examples of optimal or efficient designs for various situations. We also study some existing designs from the literature. Chapter 5 shows the details of the proofs for theorems and lemmas in the thesis.
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