Abstract A strain gradient theory of elasticity has been proposed recently to address problems involving singularities and discontinuities in materials. Among its capabilities, the theory can eliminate the crack-tip strain singularity while providing structure to the cohesive zone without resorting to extraneous forces as in plastic strip models. Due to the theory's higher-order terms, additional boundary conditions must be imposed over those of linear elasticity, which affect directly the form of the solution. Various boundary conditions have been used by different investigators resulting in different profiles in crack opening displacement. In this two-part paper the similarities and differences among the various antiplane crack solutions obtained thus far for this theory are examined. New analytical results are also presented in the course of this review. Part I of the paper is devoted to solutions having crack displacements which are oscillatory in nature; whereas, Part II is dedicated to those which exhibit monotonic behavior.