We report on a comprehensive study of the unique adhesive properties of mats of polymethylmethacrylate (PMMA) nanofibers produced by electrospinning. Fibers are deposited on glass, varying the diameter and the relative orientation of the polymer filaments (random vs aligned configuration). While no significant variation is observed in the static contact angle (about 130{deg}) of deposited water drops upon changing the average fiber diameter up to the micrometer scale, fibers are found to exhibit unequalled water adhesion. Placed vertically, they can hold up water drops as large as 60 microL, more than twice the values typically obtained with hairy surfaces prepared by different methods. For aligned fibers with anisotropic wetting behavior, the maximum volume measured in the direction perpendicular to the fibers goes up to 90 {mu}L. This work suggests new routes to tailor the wetting behavior on extended areas by nanofiber coatings, with possible applications in adsorbing and catalytic surfaces, microfluidic devices, and filtration technologies.