A discrete implicit surface is considered as the zero set of a scalar-valued function on an ambient 3-dimensional simplicial complex. In contrast to the discrete geometry of simplicial surfaces, the differential operators of implicit surfaces live on the ambient grid. The purpose of this project is to derive a discrete differential geometric toolbox for implicit surfaces similar to (explicit) discrete surfaces. This includes the development of discrete differential operators for implicit surfaces including curvature operators, and the study of curvature flows. A major application is the computation of (parametric) discrete minimal surfaces whose topology is a priori unknown.