Sub-varieties and Descent

Let $X$ be a variety and $Z$ be a sub-variety. Can one glue vector bundles on $X-Z$ with vector bundles on some small neighborhood of $Z$? We survey two recent results on the process of gluing a vector bundle on the complement of a sub-variety with a vector bundle on some 'small' neighborhood of the sub-variety. This is joint work. The first with M. Temkin and is about gluing categories of coherent sheaves over the category of coherent sheaves on a Berkovich analytic space. The second with J. Block and is about gluing dg enhancements of the derived category of coherent sheaves.