In this thesis, we describe some wall crossings in Bridgeland stability and the birational geometry of the associated moduli spaces of Bridgeland stable objects on a quartic $K3$ surface. In particular, we study the picture for objects with Chern characters $\text{Ch}(E) = (1, 0, -12)$ generically the ideal sheaf of 12 points, $\text{Ch}(E) = (1, 0, -16)$ generically the ideal sheaf of 16 points, and $\text{Ch}(E) = (1, 0, -4n^2)$ generically the ideal sheaf of $4n^2$ points. Then, we relate

more »

... se findings to studies of moduli spaces of Bridgeland stable objects in three dimensional projective space of Chern character $\text{Ch}(E) = (1, 0, -3, 5)$ generically the ideal sheaf of a twisted cubic, and to $\text{Ch}(E) = (1, 0, -4, 8)$ generically the ideal sheaf of a degree 4 elliptic curve. Finally, we examine some of the maps from the moduli spaces of Bridgeland stable objects in three dimensional projective space to the moduli spaces of Bridgeland stable objects on the $K3$ surface induced by the restriction map and relate their birational geometries.