In
mathematics, a **classification theorem** answers the classification problem "What are the objects of a given type, up to some equivalence?". It gives a non-redundant enumeration: each object is equivalent to exactly one class.

A few issues related to classification are the following.

- The equivalence problem is "given two objects, determine if they are equivalent".
- A
complete set of invariants, together with which invariants are realizable,
^{[ clarify]}solves the classification problem, and is often a step in solving it. - A computable complete set of invariants
^{[ clarify]}(together with which invariants are realizable) solves both the classification problem and the equivalence problem. - A canonical form solves the classification problem, and is more data: it not only classifies every class, but provides a distinguished (canonical) element of each class.

There exist many **classification theorems** in
mathematics, as described below.

- Classification of Euclidean plane isometries
- Classification theorems of surfaces
- Classification of two-dimensional closed manifolds
- Enriques–Kodaira classification of algebraic surfaces (complex dimension two, real dimension four)
- Nielsen–Thurston classification which characterizes homeomorphisms of a compact surface

- Thurston's eight model geometries, and the geometrization conjecture
- Berger classification
- Classification of Riemannian symmetric spaces
- Classification of 3-dimensional lens spaces
- Classification of manifolds

- Classification of finite simple groups
- Artin–Wedderburn theorem — a classification theorem for semisimple rings
- Classification of Clifford algebras
- Classification of low-dimensional real Lie algebras
- Bianchi classification
- ADE classification
- Langlands classification

- Finite-dimensional vector spaces (by dimension)
- Rank–nullity theorem (by rank and nullity)
- Structure theorem for finitely generated modules over a principal ideal domain
- Jordan normal form
- Sylvester's law of inertia

- Classification of electromagnetic fields
- Petrov classification
- Segre classification
- Wigner's classification