In mathematics, and particularly homology theory, **Steenrod's Problem** (named after mathematician Norman Steenrod) is a problem concerning the realisation of homology classes by singular manifolds.

Let *M* be a closed, oriented manifold, and let [*M*] ∈ H_{n}(*M*) be its orientation. Here H_{n}(*M*) denotes the *n*-dimensional homology group of *M*. Any continuous map ƒ : *M* → *X* defines an induced homomorphism ƒ_{*} : H_{n}(*M*) → H_{n}(*X*). A homology class of H_{n}(*X*) is called realisable if it is of the form ƒ_{*} [*M*] where [*M*] ∈ H_{n}(*M*). The Steenrod problem is concerned with describing the realisable homology classes of H_{n}(*X*).

All elements of *H*_{k}(*X*) are realisable by smooth manifolds provided *k* ≤ 6. Any elements of *H*_{n}(*X*) are realisable by a mapping of a Poincaré complex provided *n* ≠ 3. Moreover, any cycle can be realisable by the mapping of a pseudo-manifold.

The assumption that *M* be orientable can be relaxed. In the case of non-orientable manifolds, every homology class of H_{n}(*X*,**Z**_{2}), where **Z**_{2} denotes the integers modulo 2, can be realised by a non-oriented manifold ƒ : *M*^{n} → *X*.

For smooth manifolds *M* the problem reduces to finding the form of the homomorphism Ω_{n}(*X*) → H_{n}(*X*), where Ω_{n}(*X*) is the oriented bordism group of *X*. The connection between the bordisms Ω_{*} and the Thom spaces MSO(*k*) clarified the Steenrod problem by reducing it to the study of the mappings H*(MSO(*k*)) → H*(*X*). A non-realisable class, [*M*] ∈ H_{7}(*X*), has been found where *M* is the Eilenberg–MacLane space: K(**Z**_{3}⊕**Z**_{3},1).