Let $(\Omega,\mathcal{F},\mathbb{P})$ be a probability space. Let $(X_{n})_{n\in\mathbb{N}}$ be a sequence of random variables, and $X$ be another RV. Let $X_{n}\sim\mu_{n},X\sim\mu$. We say $X_n$ converges to $X$ in distribution if and only if $\forall x\in \mathbb{R}$ such that $\mu \{ x \}=0$ we have that $$\mu_{n}((-\infty,x])\to\mu((-\infty,x])$$or $$F_{n}(x)\to F(x)$$ this is also commonly referred to as convergence in law and denoted as $$X_{n}\xrightarrow{d}X.$$