The Generalized Langevin Equation (GLE) is a Stochastic Integro-Differential Equation that is commonly used to describe the velocity of microparticles that move randomly in viscoelastic fluids. Such particles commonly exhibit what is known as anomalous subdiffusion, which is to say that their position Mean-Squared Displacement (MSD) scales sublinearly with time. While it is common in the literature to observe that there is a relationship between the MSD and the memory structure of the GLE, and there exist special cases where explicit solutions exist, this connection has never been fully characterized. In this work, we first consider the unconstrained GLE in different settings and prove that large-time asymptotic behavior of the particle MSD is entirely determined by the tail behavior of the GLE's memory kernel. We then consider the GLE in the presence of a wide class of external potential wells with a special class of power-law memory. We use a Markovian system to represent the non-linear GLE and obtain results on the asymptotic properties of the solutions.