In this paper, we review the construction and large $N$ study of the continuous two-dimensional Yang--Mills theory with gauge group $\mathrm{U}(N)$ through probability, combinatorics and representation theory. In the first part, we define the continuous Yang--Mills measure using Markovian holonomy fields, following a construction by Lévy, then we show in the second part how to derive the character expansion of the partition function for any compact structure group from this setting. We continue with two developments obtained in the last few years by Dahlqvist, Lemoine, Lévy and Maïda with similar approaches with respect to the partition function: its large-$N$ asymptotics on all compact surfaces for the structure group $\mathrm{U}(N)$, and its $\frac{1}{N}$ expansion on a torus with an interpretation in terms of random surfaces.