Biological processes at the cellular level take place in heterogeneous environments, and usually involve only a small number of molecules. They tend to exhibit strong time dependent fluctuations, as a result, and are, therefore, intrinsically stochastic. The present thesis describes some of the efforts I have made during the course of my research work to develop simple, analytically tractable models of a selection of biologically-inspired problems in which this kind of stochasticity is a central ingredient. These problems are: (i) single molecule enzyme activity (ii) intermittency in single enzymes, (iii) liquids crystal dynamics (iv) modulation of electron transfer kinetics during photosynthesis, and (v) anomalous polymer translocation dynamics. All of these problems can be defined in terms of quantity that changes randomly in time because of environmental fluctuations with broad distributions of relaxation times. In this thesis I show that a generalization of a model that describes simple Brownian Motion can be used to understand many of the dynamical aspects of these problems.