Offering an architecture for social networking in which people have agency over 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 people with networked smartphones -- roaming (address-changing) computing devices communicating over an unreliable network (e.g., using UDP). The architecture incorporates (i) a decentralized social graph, where each person controls, maintains and stores only their local neighborhood in the graph; (iii) personal feeds, with authors and followers who create and store the feeds; and (ii) a grassroots dissemination protocol, in which communication among people occurs only along the edges of their social graph. The architecture realizes these components using the blocklace data structure -- a partially-ordered conflict-free counterpart of the totally-ordered conflict-based 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.