The goal of this thesis is to provide a modern interpretation and an extension of the classical works of the 1970s and 1980s constructing moduli spaces of vector bundles and coherent sheaves on projective spaces by means of "linear data", that is spaces of matrices modulo a linear group action. These works culminated with the description by Drézet and Le Potier of the moduli spaces of Gieseker-semistable sheaves on P2 as what are called today quiver moduli spaces. We show that this can be naturally understood and generalized using the language of derived categories and stability structures on them. In particular, we obtain analogous explicit constructions for moduli of sheaves on P1xP1, and we investigate these moduli spaces using the theory of quiver moduli.