Introduction to $p$-adic Langlands program for $\mathrm{GL}_2$

The $p$-adic and$\bmod p$ Langlands program is an avatar of the classical Langlands program and has been first initiated by C. Breuil. In this colloquium talk, I will give a brief introduction to the program and survey some recent progress in the case of $\mathrm{GL}_2$.