Let $F$ be a global field and $\Gamma_F$ its absolute Galois group. Given a continuous representation $\bar{\rho}: \Gamma_F \to G(k)$, where $G$ is a split reductive group and $k$ is a finite field, it is of interest to know when $\bar{\rho}$ lifts to a representation $\rho: \Gamma_F \to G(O)$, where $O$ is a complete discrete valuation ring of characteristic zero with residue field $k$. One would also like to control the local behaviour of $\rho$ at places of $F$, especially at primes dividing $p = \mathrm{char}(k)$ (if $F$ is a number field). In this talk I will give an overview of a method developed in joint work with Chandrashekhar Khare and Stefan Patrikis which allows one to construct such lifts in many cases.