Over the past decade, the development of implicit surfaces for representing three-dimensional geological structures has resulted in significant improvement in the automation and robustness of fault network modeling. However, complex fault networks and splay faults are not easily represented with implicit surfaces and call for specific modeling strategies. The standard practice in geomodeling is to represent faults as open discontinuity surfaces, without direct consideration for conceptual fault growth models. In this work, we review the origin and main features of various complex fault configurations encountered in nature. We propose three approaches combining structural geology and topology to describe faults: (1) An elementary implicit fault representation, which describes a fault by combining a series of scalar and vector fields. This approach is appropriate for isolated faults and branching faults, when using an intersection operator. (2) A piecewise explicit fault representation, which makes it possible to build faults with internal branch lines by segmenting the fault surface into several explicit charts. (3) A composite implicit fault representation, which combines several elementary faults into a complex one. This last approach is closer to structural geology and clears the path to a unified theoretical framework for the three-dimensional description of fault surfaces and their associated deformations.