A modular mounting and connection or interconnection system for microfluidic devices (20) includes a plurality of end-butting compression-sealing fluid connectors or adapters (32), and one or more clamping structures (54, 56) each structured to hold one of the fluid connectors (32) in compression against a planar surface of a microfluidic device (20), and to press against the device, on another directly opposing planar surface thereof, either a contact pad (48) or another of the fluid connectors (32), with each clamping structure (54,56) including an individually moveable compression-providing element such as a compression screw (36) structured to provide a controlled amount of compression. The system desirably further includes one or more device frames (58) each structured so as to receive and hold a microfluidic device (20) with one or more of the clamping structures (54, 56) attached, the device frame (58) being structured to retain the device (20) and attached clamping structures (54, 56) by constraining only one or two of the clamping structures (54,56) in a manner such that no torsion or bending is applied to the device (20), and one or more system frames (70) structured so as to receive and hold a plurality of device frames (58) in proximity to each other in a three-dimensional array, such that volumes of desired fluid interconnections between devices (20) can be minimized.