Let $F$ be a totally real field and $p$ be an odd prime unramified in $F$. We will give an overview of the problem of determining the explicit mod $p$ structure of a modular $p$-adic Galois representation and determining the associated local Serre weights. The Galois representations are attached to Hilbert modular forms over $F$, more precisely to eigenforms on a Shimura curve over $F$. The weight part of the Serre’s modularity conjecture for Hilbert modular forms relates the local Serre weights at a place $v|p$ to the structure of the mod $p$ Galois representation at the inertia group over $v$. Thus, local Serre weights give good information on the structure of the modular mod $p$ Galois representation. The eigenforms considered are of small slope at a fixed place $\mathbf{p}|p$, and with certain constraints on the weight over $\mathbf{p}$. This is based on a joint work with Shalini Bhattacharya.