A description is given of a generic EGSnrc Monte Carlo user code, GenUC, which was developed as an attempt to simplify and optimize the geometry and scoring coding of EGSnrc user codes. GenUC was developed using the methodology of combinatorial geometry that allows a straightforward implementation of complicated geometric setups with intersecting boundaries, where subsequent modifications to the geometry are easily performed. Presently, GenUC has five elemental volumes that can be defined in any position in space: spheres, ellipsoids, parallelepipeds, and circular cylinders and cones. The mortran macro-based implementation of the combinatorial geometry package allows an easy definition/extension of any other elemental volume, e.g., elliptical cylinders and cones. The scoring of the relevant parameters and the output of the results in GenUC are performed with two CERN data analysis packages, which permit the generation of nonplanar phase space distribution files and can also be used for geometry verification, among many other capabilities. GenUC has been successfully applied to complex geometric setups, e.g., intracavitary brachytherapy applicators and was also benchmarked against the EGSnrc user code, DOSRZnrc.