This paper describes a procedure for making topologically complex three-dimensional microfluidic channel systems in poly(dimethylsiloxane) (PDMS). This procedure is called the "membrane sandwich" method to suggest the structure of the final system: a thin membrane having channel structures molded on each face (and with connections between the faces) sandwiched between two thicker, flat slabs that provide structural support. Two "masters" are fabricated by rapid prototyping using two-level photolithography and replica molding. They are aligned face to face, under pressure, with PDMS prepolymer between them. The PDMS is cured thermally. The masters have complementary alignment tracks, so registration is straightforward. The resulting, thin PDMS membrane can be transferred and sealed to another membrane or slab of PDMS by a sequence of steps in which the two masters are removed one at a time; these steps take place without distortion of the features. This method can fabricate a membrane containing a channel that crosses over and under itself, but does not intersect itself and, therefore, can be fabricated in the form of any knot. It follows that this method can generate topologically complex microfluidic systems; this capability is demonstrated by the fabrication of a "basketweave" structure. By filling the channels and removing the membrane, complex microstructures can be made. Stacking and sealing more than one membrane allows even more complicated geometries than are possible in one membrane. A square coiled channel that surrounds, but does not connect to, a straight channel illustrates this type of complexity.