Using adaptive optics on the Keck II 10 m telescope on Mauna Kea, we have surveyed 24 of the nearest young stars known in search of close companions. Our sample includes members of the MBM 12 and TW Hydrae young associations and the classical T Tauri binary UY Aurigae in the Taurus star-forming region. We present relative photometry and accurate astrometry for 10 close multiple systems. The multiplicity frequency in the TW Hydrae and MBM 12 groups are high in comparison to other young regions, although the significance of this result is low because of the small number statistics. We resolve S18 into a triple system, including a tight 63 mas (projected separation of 17 AU at a distance of 275 pc) binary, for the first time, with a hierarchical configuration reminiscent of VW Chamaeleontis and T Tauri. Another tight binary in our sample—TWA 5Aab (54 mas or 3 AU at 55 pc)—offers the prospect of dynamical mass measurement using astrometric observations within a few years and thus could be important for testing pre–main-sequence evolutionary models. Our observations confirm with 9 σ confidence that the brown dwarf TWA 5B is bound to TWA 5A. We find that the flux ratio of UY Aur has changed dramatically, by more than a magnitude in the H band, possibly as a result of variable extinction. With the smaller flux difference, the system may once again become detectable as an optical binary, as it was at the time of its discovery in 1944. Taken together, our results demonstrate that adaptive optics on large telescopes is a powerful tool for detecting tight companions and thus exploring the frequency and configurations of close multiple systems.