The present invention is directed to nonwoven webs and articles comprising nanofibers. The nanofibers are made from a polymer having a glass transition temperature about 25 C. The nanofibers, having a diameter of less than 1 micron, may comprise a significant number of the fibers in one layer of the web contained by the article. Preferably, the nanofibers are produced in a melt film fibrillation process. Articles include diapers, training pants, adult incontinence pads, catamenials products such as feminine care pads and pantiliners, tampons, personal cleansing articles, personal care articles, and personal care wipes including baby wipes, facial wipes, and feminine wipes.