Let M 0 3 M_0^3 and M 1 3 M_1^3 be compact, oriented 3 3 -manifolds. They are homology cobordant (respectively relative homology cobordant) if ∂ M 1 3 = ∅ ( resp. ∂ M 1 3 ≠ ∅ ) \partial M_1^3 = \emptyset \;({\text {resp.}}\;\partial M_1^3 \ne \emptyset ) and there is a smooth, compact oriented 4 4 -manifold W 4 {W^4} such that ∂ W 4 = M 0 3 − M 1 3 \partial {W^4} = M_0^3 - M_1^3 (resp. ∂ W 4 = M 0 3 − M 1 3 ) ∪ ( M i 3 × [ 0 , 1 ] ) \partial {W^4} = M_0^3 - M_1^3) \cup (M_i^3 \times [0,1]) and H ∗ ( M i 3 ; Z ) → H ∗ ( W 4 ; Z ) {H_{\ast }}(M_i^3;{\mathbf {Z}}) \to {H_{\ast }}({W^4};{\mathbf {Z}}) are isomorphisms, i = 0 , 1 i = 0,1 . Theorem. Every closed, oriented 3 3 -manifold is homology cobordant to a hyperbolic 3 3 -manifold. Theorem. Every compact, oriented 3 3 -manifold whose boundary is nonempty and contains no 2 2 -spheres is relative homology cobordant to a hyperbolic 3 3 -manifold. Two oriented links L 0 {L_0} and L 1 {L_1} in a 3 3 -manifold M 3 {M^3} are concordant if there is a set A 2 {A^2} of smooth, disjoint, oriented annuli in M × [ 0 , 1 ] M \times [0,1] such that ∂ A 2 = L 0 − L 1 \partial {A^2} = {L_0} - {L_1} , where L i ⊆ M 3 × { i } , i = 0 , 1 {L_{i}} \subseteq \;{M^3} \times \{ i\} ,i = 0,1 . Theorem. Every link in a compact, oriented 3 3 -manifold M 3 {M^3} whose boundary contains no 2 2 -spheres is concordant to a link whose exterior is hyperbolic. Corollary. Every knot in S 3 {S^3} is concordant to a knot whose exterior is hyperbolic.