Gaussian graphical models are used throughout the natural sciences, social sciences, and economics to model the statistical relationships between variables of interest in the form of a graph. We here provide a pedagogic introduction to Gaussian graphical models and review recent results on maximum likelihood estimation for such models. Throughout, we highlight the rich algebraic and geometric properties of Gaussian graphical models and explain how these properties relate to convex optimization and ultimately result in insights on the existence of the maximum likelihood estimator (MLE) and algorithms for computing the MLE.