The classical Jacquet-Langlands (J-L) correspondence relates complex representations of $\mathrm{GL}(n)$ and that of its inner forms. It was proved in 1970’s and provides one of the first examples of the functoriality conjecture in Langlands program. However, when we consider representations with $p$-adic or mod $p$ coefficients, the analogue of J-L correspondence is still poorly understood, even in the simplest case of $\mathrm{GL}(2,\mathbb{Q}_p)$. In this talk, we will report some recent progress on the $p$-adic and mod $p$ J-L correspondence for$\mathrm{GL}(2,\mathbb{Q}_p)$. This is joint work with Haoran Wang.