`Heterotic Orbifold and Resolution Model Building' Abstract: We consider four dimensional theories that arise as compactifications of the heterotic string on toroidal orbifolds and their blowup counterparts. The first one is formulated as an exactly solvable conformal worldsheet theory, whereas the second ones are only accessible in the SUGRA approximation. Although these descriptions are different, these theories seem to live in the same moduli space and are thus corners of one underlying theory. In addition, we find that gauged linear sigma models provide a framework that is able to interpolate between them.