I will motivate and provide an overview of recent efforts in my research group on the design and analysis of stochastic-gradient-based algorithms for solving constrained optimization problems. I will also share more detailed looks at two recent projects, one on the almost-sure convergence of primal and dual iterates generated by one such algorithm, and another on active-set identification by noisy and stochastic optimization algorithms. Identifying the constraints that are active at a solution of an optimization problem is important both theoretically and practically, such as for certifying optimality and sensitivity analysis. I will show how state-of-the-art identification techniques can be extended from deterministic to noisy and stochastic settings, and demonstrate our results with a constrained supervised learning problem.