Kuznetsov components of Fano threefolds

Appears in collection : Géométrie algébrique en l'honneur de Claire Voisin

I will give an overview of recent results on Kuznetsov components of Fano threefolds. I will emphasize how they help organise the classification and moduli spaces of prime Fano threefolds. The results are based on Bridgeland stability conditions, Brill-Noether loci, and enhanced equivariant categories.