The purpose of this thesis, as implied by its title, is essentially two-fold. The first goal is to introduce and study the notion of properness, as well as some other concepts related to it. The importance of this property lies in the fact that in the infinite-dimensional case it effectively compensates for the lack of interior points of the positive cone of a commodity space. Thus, in the infinite-dimensional setting, properness allows to resort to the familiar notion of prices supporting equilibria, a key ingredient in proving that an equilibrium does (or does not) exist. The second major objective is to investigate necessary and sufficient conditions for local and uniform properness of von Neumann-Morgenstern (separable) utility functions. These functions, normally defined on the positive cone of some Lp (μ), where 1 ≤ p ≤ ∞, play an important role in economics and financial applications. While studying the uniform properness of certain separable utility functions, we dispose of the requirement that their kernel is differentiable and prove some elegant results that are new to the literature. We also give an original example demonstrating a fundamental difference between Lp and lp for the purpose of establishing uniform properness of separable utility functions in these spaces. While working towards the above two goals, we also provide the reader with an overview of the modern literature on equilibrium theory. In particular, we highlight one interesting recent development known as limited arbitrage, and illustrate it with examples suggesting that this concept is not always self-consistent. Finally, we apply the theory developed throughout the rest of the thesis to establish necessary and sufficient conditions for the existence of an equilibrium in security markets. We extend our findings to a model of financial market where traders' utility functions are separable, thus obtaining a new version of the existence result.