We study the geometry and topology of Quot schemes on smooth projective curves. First, we give an explicit presentation of the rational cohomology ring of the Quot scheme parametrising torsion quotients on $\PP^1$. Next, we construct a stratification of the corresponding relative Quot scheme, which recovers several known results by specialisation. We also use this stratification to prove that the integral cohomology of the Quot scheme parametrising torsion quotients is torsion-free, thereby strengthening the first result. Finally, we study the cohomology of Schur complexes associated with tautological complexes on the Quot scheme parametrising positive rank quotients on $\PP^1$, and we construct exceptional collections in its derived category.
