Systems biology is located at the intersection of biology, computer science, and mathematics, and is based on the translation of biological systems into mathematical models. It aims to predict the behavior of these biological systems to improve the efficiency of time- and cost-intensive research in laboratories. In this thesis, we focus on the description and understanding of metabolic networks at steady state. These networks are mathematical models of the metabolic processes inside a cell. The stoichiometric and thermodynamic constraints that must hold in a metabolic network at steady-state define the steady-state flux cone. An im- portant concept to analyze flux cones in a mathematically and biologically meaningful way are elementary flux modes, which can be considered as mini- mal functional units of metabolic networks. In the flux cone, elementary flux modes correspond to vectors with inclusionwise minimal support. We focus on geometric aspects of flux cones of metabolic networks and elementary flux modes. The number of elementary flux modes may be very large, even for medium-sized metabolic networks. We study the facial structure and investigate the distribution of elementary flux modes among the faces of the flux cone. We observe that they are primarily contained in faces of relatively low dimension. Due to this observation, we develop a method to enumerate subsets of elementary flux modes that are contained in a specific face of the flux cone and apply this to decompositions of flux vectors. Empirically, we observed that elementary flux modes can always be written as a positive sum of exactly two others. Motivated by this, we investigate decompositions of elementary flux modes into others and discuss a conjecture that claims each EFM can always be decomposed into exactly 2 others or is not decomposable at all. Our mathematical results are illustrated on real examples and the presented data can be reproduced with a Python package we developed.