Selberg's central limit theorem is an influential probabilistic result in analytic number theory which roughly states that the logarithm of the Riemann zeta-function $\zeta(s)$ on the half-line, that is $\Re s = \frac12$, has an approximate two-dimensional Gaussian distribution as $\Im s \to \infty$. We will carefully review the important ideas in the proof of Selberg's theorem and then will mention some variants of it. Towards the end of the talk, we will also see some of its applications.