The weight part of Serre's conjecture

Serre's conjecture (now a theorem) predicts that an irreducible 2-dimensional odd
Galois representation of $\mathbb Q$ with coefficients in $\bar{\mathbb F}_p$ comes from the mod p reduction of
a modular form. A key feature is that two modular forms of different weights can have the same
mod p reduction. Fixing a modular form $f$, the weight part of Serre's conjecture seeks to find all
the possible weights where one can find a modular form congruent to $f$ mod $p$. The recipe for these
weights was conjectured by Serre, and it depends only on the local Galois representation at $p$. I
will explain the ideas involved in Edixhoven's proof of the weight part, and if time allows, I
will briefly say something about what the generalizations beyond $\operatorname{GL}_2/\mathbb Q$ might look like.