Mixture approximations are relevant to many problems in science and engineering, including distributed time delays in differential equations and stochastic differential equations, for which the solution is a time-varying probability distribution. In this presentation, we will discuss Gaussian, Erlang, and beta mixture approximations of density-like functions with infinite, semi-infinite, and bounded support, respectively. In particular, we will discuss how Erlang and beta approximations can be used to simplify the analysis and simulation of delay differential equations with distributed time delays by permitting a transformation to a set of ordinary differential equations (ODEs). Specifically, we will demonstrate its utility for steady state stability analysis and delay identification, and we will discuss the perspectives of using it in controller design, e.g., PID controllers with forgetting factors and delay-adaptive control.