The Strominger system is a system of partial differential equations describing the geometry of compactifications of heterotic superstrings with flux. Mathematically it can be viewed as a generalization of Ricci-flat metrics on non-Kshler Calabi-Yau 3- folds. In this thesis, I will present some explicit solutions to the Strominger system on a class of noncompact Calabi-Yau 3-folds. These spaces include the important local models like C3 as well as both deformed and resolved conifolds. Along the way, I also give a new construction of non-Kihler Calabi-Yau 3-folds and prove a few results in complex geometry. Advisors/Committee Members: Shing-Tung Yau, Victor Guillemin (advisor).