A common property of many large networks, including the Internet, is that the connectivity of the various nodes follows a scale-free power-law distribution, P(k) = ck(-alpha). We study the stability of such networks with respect to crashes, such as random removal of sites. Our approach, based on percolation theory, leads to a general condition for the critical fraction of nodes, p(c), that needs to be removed before the network disintegrates. We show analytically and numerically that for alpha</=3 the transition never takes place, unless the network is finite. In the special case of the physical structure of the Internet (alpha approximately 2.5), we find that it is impressively robust, with p(c)>0.99.