The aim of this work is to explore the curvature properties of one-relator groups and investigate the conjecture that every torsion free one-relator group occurs as the fundamental group of a combinatorial 2-complex with the nonpositive immersions property, a property we define in Chapter 1. The theory of conformal non-positive curvature is central to our strategy for approaching this conjecture. Chapter 1 introduces the idea of conformal non-positive curvature and gives a proof of Wise’s theorem that non-positive sectional curvature implies the non-positive immersions property. Our proof introduces some intermediate properties which then get used in Chapter 5 to extend Wise’s theorem to apply to complexes with 2-sectional curvature. Our strategy for proving that a group has a complex with non-positive sectional curvature is to deform the standard complex for the group by homotopy equivalences and check the resulting complex for a suitable angle assignment. To determine how to deform the standard complex, we turn to the theory of special complexes. Chapter 2 presents some established techniques from the field of computational 3-manifolds for constructing special complexes. Also in Chapter 2, we introduce a method called drawing the link which is better suited to deforming an arbitrary complex to make it special. In Chapter 4, we describe an algorithm which uses concepts and techniques from Chapter 2 to deform an arbitrary complex to make it special. We also give a proof of termination of the algorithm and show results of running our implementation of our algorithm on a large collection of complexes for two-generator one-relator groups. In Chapter 3, we present a series of results aimed at simplifying the problem of proving that one-relator groups are conformally non-positively curved. We appeal to a theorem of classical one-relator group theory to say that it is sufficient to prove the two-generator case, and we show how for a special complex for one of these groups, the problem of determining whether non-positive sectional curvature can be attained reduces to a nice linear programming system. Also in Chapter 3, we prove a collection of theorems for quickly identifying complexes with non-positive sectional curvature and with the non-positive immersions property. Key among these results is a theorem stating that the non-positive immersions property is preserved by the attaching of a surface with boundary provided that the attaching maps are π ₁-injective. This theorem allows for the construction of many non-trivial examples that demonstrate that the converse of Wise’s theorem is false. The final theorem in Chapter 5 combines ideas from all previous chapters. It demonstrates that the non-positive 2-sectional curvature property, when present, is always exhibited by a standard 2-complex.