The radially engaging coupling system comprises numerous embodiments for connecting two or more concentric components with radially extending or retracting bolts. Most embodiments include a ring having a plurality of Archimedean spiral slotted segments. Each bolt has a pin extending laterally therefrom to engage a corresponding slot in the ring. Rotation of the ring drives the pins and their bolts outward or inward to engage or disengage an outer component. Some embodiments have inwardly or outwardly spaced pin passages in the ring, rather than the spiral slots. The disc may comprise a snap ring. Another embodiment has two axially spaced spirally slotted rings, with a plurality of cylindrical rollers therebetween. Rotation of the rings drives the rollers inward or outward to engage corresponding axial grooves in the inner shaft. In another embodiment each bolt has an arcuate segment extending thereacross, to engage a corresponding circumferential groove in an outer component.