This presentation explores Mean Field Games (MFGs) through the lens of functional analysis, focusing on the role of monotonicity methods in understanding their properties. We begin by introducing MFGs as models of large populations of interacting rational agents and illustrate their derivation for deterministic problems. We then examine key questions regarding the existence and uniqueness of MFG solutions. Specifically, we present several new existence theorems obtained via $p$-Laplacian regularization. Finally, we discuss weak-strong uniqueness and establish conditions under which weak and strong solutions of MFGs coincide.