Busemann's theorem states that the intersection body of an origin-symmetric convex body is also convex. We provide a version of Busemann's theorem for p-convex bodies. We show that the intersection body of a p-convex body is q-convex for certain q. Furthermore, we discuss the sharpness of the previous result by constructing an appropriate example. Finally, we extend these theorems to some general measure spaces with log-concave and s-concave measures.