This dissertation is composed of two parts, a theoretical part, in which certain asymptotic properties of near critical branching processes are studied, and an applied part, consisting of statistical analysis of cell growth data. First, near critical single type Bienaymé-Galton-Watson (BGW) processes are considered. It is shown that, under appropriate conditions, Yaglom distributions of suitably scaled BGW processes converge to that of the corresponding diffusion approximation. Convergences of stationary distributions for <italic>Q</italic> processes and models with immigration to the corresponding distributions of the associated diffusion approximations are established as well. Moreover, convergence of Yaglom distributions of suitably scaled multitype subcritical BGW processes to that of the associated diffusion model is established. Next, near critical catalyst-reactant branching processes with controlled immigration are considered. The catalyst population evolves according to a classical continuous time branching process, while the reactant population evolves according to a branching process whose branching rate is proportional to the total mass of the catalyst. Immigration takes place exactly when the catalyst population falls below a certain threshold, in which case the population is instantaneously replenished to the threshold. A diffusion limit theorem for the scaled processes is established, in which the catalyst limit is a reflected diffusion and the reactant limit is a diffusion with coefficients depending on the reactant. Stochastic averaging under fast catalyst dynamics are considered next. In the setting where both catalyst and reactant evolve according to the above described (reflected) diffusions, but where the evolution of the catalyst is accelerated by a factor of <italic>n</italic>, we establish a scaling limit theorem, in which the reactant process is asymptotically described through a one dimensional SDE with coefficients depending on the invariant distribution of the catalyst reflected diffusion. Convergence of the stationary distribution of the scaled catalyst branching process (with immigration) to that of the limit reflected diffusion is established as well. Finally, results from a collaborative proof-of-principle study, relating cell growth to the stiffness of the surrounding environment, are presented.