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Entwined structures (A,C,ψ) were introduced by Brzeziński and Majid to study the interdependence of an R-algebra A and an R-coalgebra C, R a commutative ring. It turned out that this relationship can also be expressed by the fact that A⊗_RC has a canonical A-coring structure. More generally, weak entwined structures and their modules were studied by Caenepeel and de Groot and Caenepeel suggested relating these to pre-corings. Slightly modifying this notion we introduce weak corings and develop a general theory of comodules over such corings. In particular we obtain that (A,C,ψ) is a weak entwined structure if and only if A⊗_RC is a weak A-coring (with canonical structure maps). Weak bialgebras in the sense of Böhm–Nill–Szlachányi are characterized as R-modules with an algebra and coalgebra structure (B,μ,Δ) such that B⊗_RB is a weak coring for the various coring structures induced by μ, μ∘τ, Δ, and τ …