This thesis is divided into two parts. Chapters 2-5 describe work on the statistical physics of cellular membranes, motivated by experiments that suggest they are tuned close to a two dimensional liquid-liquid critical point. Chapter 6 describes work towards an information theoretic understanding of how simple effective descriptions emerge out of systems with complicated microscopic details. Chapter 1 gives a detailed introduction to both of these topics. Chapter 2 presents a minimal model for a cellular membrane consisting of a nearly critical two dimensional fluid coupled to a fixed cortical cytoskeleton. We argue that proximity to criticality is thermodynamically necessary to explain the presence of heterogeneity at 10 [-] 100nm, as is commonly observed in experiments. We further show that this model naturally recapitulates many of the findings in the membrane 'raft' literature. In chapter 3 we argue that proximity to criticality in the membrane is distinguished, in part, by the presence of long ranged critical Casimir forces that act between membrane bound proteins. We estimate the form of this potential using several techniques, and show that it is expected to be ~ kB T over tens of nanometers. We further argue that these forces could be playing important roles in cellular processes. In chapter 4 we show that the dynamics of synthetic membranes tuned close to a critical point are in a newly predicted universality class particular to two dimensional liquids immersed in a three dimensional, non-critical, bulk fluid. With just a single free parameter, a model for this universality class quantitatively describes all of the observable time dependent correlation functions. In chapter 5 we explore the possibility that general anesthetics act by taking the membrane away from its liquid-liquid critical point. We present experimental evidence that shows that general anesthetics do indeed depress the critical temperature (T c ) in cell-derived vesicles by approximately 4K. In addition, we show that a receptor allosterically regulated by the membrane's composition could be sufficiently disrupted by this change in T c to explain the most relevant phenomenology of anesthesia- that certain ligand-gated ion channels have their response to ligand dramatically potentiated. In chapter 6 we apply an information theoretic framework to two models from statistical physics, where we see the emergence of a continuum description of diffusion and of the universal behavior seen at the Ising critical point. As these develop, we find that a characteristic hierarchy of parameter importance emerges, similar to that seen in 'sloppy' models from systems biology and elsewhere.