I will develop and analyze stochastic models for two important processes in cellular biophysics. The first problem concerns mRNA translation and protein production, and is modeled as an interacting particle system in 1D. The effects of "slow codons," or defects in the mRNA, on protein production rates are addressed by asymptotic matching of mean-field solutions of the problem. In the second problem, a stochastic model for viral entry into cells is developed. The entry of viruses turns out to be a competition between membrane fusion and endocytosis. The probabilities for entry via each of these pathways are calculated within one- and two-surface receptor models. Conditions for endocytosis are mapped. Time permitting, I will also briefly introduce a stochastic inverse problem where transition rates of a Markov process can or cannot be reconstructed from first passage time distributions.