Abstract: This invention provides absorbent products, having a permeable cover made of a microperforated fibrillated thermoplastic film that has improved properties such as intimate, but nonoccluding contact with the body, yet providing comfort and resilience and immediate fluid acceptance and distribution. The cover is characterized by having non perforate regions and perforate regions, the perforate regions having a plurality of microholes defined by a network of fiberlike elements, the perforate regions being, at least in part, at elevations that are higher than the nonperforate regions. The perforate regions are caused to be at higher elevations by methods such as tensioning the cover or by partly adhering the cover to the absorbent body of the product or by combinations thereof. The microholes may be of different sizes and shapes.