How do networks form and what is their ultimate topology? Most of the literature that addresses these questions assumes complete information: agents know in advance the value of linking to other agents, even with agents they have never met and with whom they have had no previous interaction (direct or indirect). This paper addresses the same questions under what seems to us to be the much more natural assumption of incomplete information: agents do not know in advance -- but must learn -- the value of linking to agents they have never met. We show that the assumption of incomplete information has profound implications for the process of network formation and the topology of networks that ultimately form. Under complete information, the networks that form and are stable typically have a star, wheel or core-periphery form, with high-value agents in the core. Under incomplete information, the presence of positive externalities (the value of indirect links) implies that a much wider collection of network topologies can emerge and be stable. Moreover, even when the topologies that emerge are the same, the locations of agents can be very different. For instance, when information is incomplete, it is possible for a hub-and-spokes network with a low-value agent in the center to form and endure permanently: an agent can achieve a central position purely as the result of chance rather than as the result of merit. Perhaps even more strikingly: when information is incomplete, a connected network could form and persist even if, when information were complete, no links would ever form, so that the final form would be a totally disconnected network. All of this can occur even in settings where agents eventually learn everything so that information, although initially incomplete, eventually becomes complete.