Bridgeland stability is notoriously hard to construct on varieties of dimension > 2. One promising workaround is to focus on the Kuznetsov component of Fano varieties — a residual piece of the derived category that often captures the essential geometric information. In this talk, I will explain the construction of stability conditions on Kuznetsov components using the machinery of quadric fibrations introduced by Bayer–Lahoz–Macrì–Stellari. I will illustrate how this technique applies to cubic 5-folds as a new higher-dimensional example. In the second part, I will then describe properties of the associated moduli spaces. In particular, they admit immersions into hyper-Kähler varieties with Lagrangian image, generalizing a geometric construction of Illiev–Manivel.