New functional representations are needed for describing aspheric surfaces that can compensate for a high degree of wavefront asphericity and represent steeply sloped surfaces as the surface normal becomes perpendicular to the optical axis. The explicit definition of the standard aspheric surface limits the range of surfaces that it can properly describe. This paper presents both a parametrically defined surface approach and an implicitly defined surface approach. The utility of these novel approaches is demonstrated using examples of current interest. Additionally, this work shows that a Fourier series is not a useful optical surface.