Offering an architecture for social networking in which the members are in control of their personal information and social graph is an open challenge. Here we present a grassroots architecture for serverless, permissionless, peer-to-peer social networks termed Grassroots Social Networking that aims to address this challenge. The architecture is geared for roaming (address-changing) agents communicating over an unreliable network, e.g., smartphones communicating via UDP. The architecture incorporates (i) a decentralized social graph, where each member controls, maintains and stores only their local neighborhood in the graph; (ii) member-created feeds, with authors and followers who create and store the feeds; and (iii) a grassroots dissemination protocol, in which communication among members occurs only along the edges of the social graph. The architecture realizes these components using the blocklace data structure -- a distributed partially-ordered counterpart of the replicated totally-ordered blockchain. We provide two example Grassroots Social Networking protocols -- Twitter-like and WhatsApp-like -- and address their security (safety, liveness and privacy), spam/bot/deep-fake resistance, and implementation, demonstrating how server-based social networks could be supplanted by a grassroots architecture.