This paper describes a method for fabricating three-dimensional (3D) microfluidic channel systems in poly(dimethylsiloxane) (PDMS) with complex topologies and geometries that include a knot, a spiral channel, a "basketweave" of channels, a chaotic advective mixer, a system with "braided" channels, and a 3D grid of channels. Pseudo-3D channels, which are topologically equivalent to planar channels, are generated by bending corresponding planar channels in PDMS out of the plane into 3D shapes. True 3D channel systems are formed on the basis of the strategy of decomposing these complex networks into substructures that are planar or pseudo-3D. A methodology is developed that connects these planar and/or pseudo-3D structures to generate PDMS channel systems with the original 3D geometry. This technique of joining separate channel structures can also be used to create channel systems in PDMS over large areas by connecting features on different substrates. The channels can be used as templates to form 3D structures in other materials.