Let $B$ be a Borel subgroup in a reductive algebraic group $G$ over a field $k$. We study the cohomology $H^\bullet(B, \lambda)$ of 1-dimensional $B$-modules $\lambda$. When $\mathrm{char} k = 0$ there is an easy a well-known description of this cohomology whereas the corresponding problem in characteristic $p>0$ is wide open. We develop some new techniques which enable us to calculate all such cohomology in degrees at most 3 when $p$ is larger than the Coxeter number for $G$. Our methods also apply to the corresponding question for quantum groups at roots of unity.