We introduce and study the notion of filling links in 3-manifolds: a link L is filling in M if for any 1-spine G of M which is disjoint from L, π₁(G) injects into π₁(M∖L). A weaker "k-filling" version concerns injectivity modulo k-th term of the lower central series. For each k>1 we construct a k-filling link in the 3-torus. The proof relies on an extension of the Stallings theorem which may be of independent interest. We discuss notions related to "filling" links in 3-manifolds, and formulate several open problems. The appendix by C. Leininger and A. Reid establishes the existence of a filling hyperbolic link in any closed orientable 3-manifold with π₁(M) of rank 2.