The purpose of this dissertation is to study the strong minimum/maximum principle and to investigate various Liouville-type theorems for partial trace equa tions with nonlinear gradient terms. To investigate the problems and obtain the results presented in this dissertation, we will use various principles (minimum and maximum principles, a comparison principle for viscosity sub/supersolutions, boundary principles, and compact support principles), concepts of duality, and different mathematical estimates and inequalities. In this study, we examine a strong minimum principle of Vázquez type for partial trace operators with gradient terms. More explicitly, given a n-tuple a = (a1, · · · ,an) of non-negative real numbers with an > 0, we give sufficient conditions on a continuous function h: R × R+ 0 → R in order for non-negative viscosity supersolutions of Pa(D2u) = h(u,|Du|) (⋆) in connected open subsets of Rn that vanish at some point to Ω vanish identically in Ω. When h depends only on the gradient, the condition is also necessary. Here Pa belongs to a class of fully nonlinear degenerate elliptic operators that includes the Min-Max operator which is defined as the sum of the minimum and the maximum eigenvalues of the Hessian matrix. Under suitable conditions on h and a = (a1,··· ,an), both the Strong Maximum Principle and the Compact Support Principle for subsolutions are also investigated. The work covers a new class of degenerate operators and a wide class of Hamiltonians not investigated in the literature and some of the results are new even when Pa reduces to the standard Laplacian. Illustrative examples are presented for such equations. We investigate various Liouville-type theorems for partial trace equations with nonlinear gradient terms. Specifically, we establish sufficient conditions for whic