We can think of a map to projective space as a line bundle together with sections that don't vanish all together at a point. A quasi-map is just a line bundle with sections. After imposing different stability conditions on the quasi-map we get compactifications of maps to projective space different from the Kontsevich moduli of stable maps. We will discuss some wall crossing phenomena between these spaces. The main reference is Young-Hoon Kiem's paper with the same title.