This thesis concerns the asymptotic growth of solutions to nonlinear functional differential equations, both random and deterministic. How quickly do solutions grow? How do growth rates of solutions depend on the memory and the nonlinearity of the system? What is the effect of randomness on the growth rates of solutions? We address these questions for classes of nonlinear functional differential equations, principally convolution Volterra equations of the second kind. We first study deterministic equations with sublinear nonlinearity and integrable kernels. For such systems, we prove that the growth rates of solutions are independent of the distribution of the memory. Hence we conjecture that stronger memory dependence is needed to generate growth rates which depend meaningfully on the delay structure. Using the theory of regular variation, we then demonstrate that solutions to a class of sublinear Volterra equations with non–integrable kernels grow at a memory dependent rate. We complete our treatment of sublinear equations by examining the impact of stochastic perturbations on our previous results; we consider the illustrative and important cases of Brownian and alpha–stable Lévy noise. In summary, if an appropriate functional of the forcing term has a limit L at infinity, solutions behave asymptotically like the underlying unforced equation when L = 0 and like the forcing term when L is infinite. Solutions inherit properties of both the forcing term and underlying unforced equation for finite and positive L. Similarly, we prove linear discrete Volterra equations with summable kernels inherit the behaviour of unbounded perturbations, random or deterministic. Finally, we consider Volterra integro–differential equations with superlinear nonlinearity and nonsingular kernels. We provide sharp estimates on the rate of blow–up if solutions are explosive, or unbounded growth if solutions are global. We also recover well–known necessary and sufficient conditions for finite–time blow–up via new methods.