In this chapter, we start by introducing various notions of convergence for measurable functions on a measure space \((\Omega , \mathcal{A},\mu )\) . Pointwise convergence is a familiar notion of convergence for sequences of functions. However, this does not take into account the underlying measure space structure. We introduce and study notions of convergence which are more specific to and relevant in the context of sequences of measurable functions on a measure space. After a quick introduction to these notions of convergence on a general measure space, we specialize to the extremely important topic of various notions of convergence for random variables. We study this in great detail and as a natural culmination of that, we finally delve into two of the classical limit theorems in probability, popularly known as the Laws of Large Numbers. At the end, we are also going to have a brief discussion on notions of convergence for random vectors.