Abstract: This will be a review talk concerning notions of stability and particular descriptions of vector bundles on some classes of algebraic varieties.

In particular, we will sketch the main ideas appearing in Stability restriction theorem, Narasimhan-Seshadri theorem and try to review the accessible descriptions of some vector bundles, i.e. Barth construction and Horroks-Mumford bundle (indecomposible rank 2 bundle on a $\mathbb{P}^4$).

After that we will focus on some possible descriptions of moduli spaces and show how to try to prove stability of vector bundles for some particular examples. If time permits, we will probably discuss something about the metric.