A ballistics vest of the soft body armor type comprises multiple overlying layers of a thin, flexible fabric made of woven plastic fibers secured together to form a soft, flexible front panel located on a strike side of the vest, and multiple overlying layers of a thin, flexible imperforate plastic sheet stacked behind the flexible front panel, on a body side of the vest. The flexible plastic sheets preferably comprise an array of plastic fibers embedded in a resinous matrix that forms the sheet material. The flexible front panel and the stack of flexible plastic sheets behind it have a combined areal weight not greater than about 1.20 lbs/ft.sup.2 and have an NIJ Standard maximum backface of about 44 mm with a ballistics resistance that successfully prevents projectile penetration of the combined stacks of flexible layers according to NIJ Standard 0101.03 for Threat Level IIIA. The flexible fabric layers in the front panel are preferably quilted in a one-inch quilt pattern, and the fibers contained in the front panel and the rear panel are preferably extended chain polyethylene fibers. In one embodiment, the vest contains 20 layers of the woven fabric and 23 layers of the flexible plastic sheet material, with the areal density of the sheet material being 4.5 oz/yd.sup.2 and the fibers in both front and rear panels having a fiber tenacity of at least about 35 gm/denier. Other embodiments comprise extremely lightweight ballistic vests including one meeting Threat Level II specifications and having an areal weight of less than one pound per square foot (psf); another vest meets Threat Level IIA specifications with an areal weight below 0.9 psf.