Smooth cubic hypersurfaces $X\subset \mathbb{P}^{5}$ (over $\mathbb{C}$) are linked to K3 surfaces via their Hodge structures, due to the work of Hassett, and via a subcategory ${\mathcal{A}}_{X}\subset \text{D}^{\text{b}}(X)$, due to the work of Kuznetsov. The relation between these two viewpoints has recently been elucidated by Addington and Thomas. In this paper, both aspects are studied further and extended to twisted K3 surfaces, which in particular allows us to determine the group of autoequivalences of ${\mathcal{A}}_{X}$ for the general cubic fourfold. Furthermore, we prove finiteness results for cubics with equivalent K3 categories and study periods of cubics in terms of generalized K3 surfaces.

Allcock, Carlson and Toledo defined a period map for cubic threefolds which takes values in a ball quotient of dimension 10. A theorem of Voisin implies that this is an open embedding. We determine its image and show that on the algebraic level this amounts to identification of the algebra of $\operatorname{SL}(5,\mathbb{C})$-invariant polynomials on the representation space $\operatorname{Sym}^3(\mathbb{C}^5)^*$ with an explicitly described algebra of meromorphic automorphic forms on the complex 10-ball.