Reflected Brownian motion in orthants is a fundamental stochastic process with important connections to queueing theory and interacting particle systems. In this talk, we first introduce the probabilistic framework used to define such processes, together with the main results concerning recurrence and transience. We then explain how invariant measures, Green functions, and escape or absorption probabilities can be studied using kernel functional equations and analytic methods. This approach also provides access to the associated Martin boundary and to harmonic functions.