Abstract: This talk is an overview of a genre of theorems called "Grothendieck-Lefschetz theorems". I will start by introducing Lefschetz's original theorem. Our next, stop is Grothendieck's theorem for divisors and line bundles which can be though of as a codimension one Lefschetz theorem in the algebraic category.

From this point there are two natural paths to generalisation. The first is to higher codimension subvarieties. In this direction there are some infinitesimal results due to Girivaru-Patel.

The second path considers higher rank bundles. The first results in this direction are due to Hartshorne. In joint work with R. Girivaru, Hartshorne's work has been generalised to principal bundles. The talk will end with some applications of our result to splitting criteria.