Abstract We provide some explicit examples of nonlocal s-minimal surfaces for 0<s<1, namely stationary for the s-perimeter functional, introduced first in the form of a singular integral, related to fractional Laplacian, by Caffarelli, Roquejoffre and Savin, which recovers the classical concept in the limit s-->1. We find and analyze stability of a class of Lawson cones which shows important differences with the classical case. In addition we construct an axially symmetric analog of the catenoid in R3 in the limiting case when s approaches 1.