Classification theorem

• The equivalence problem is "given two objects, determine if they are equivalent".
• A complete set of invariants, together with which invariants are realizable, solves the classification problem, and is often a step in solving it.
• A computable complete set of invariants 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 element of each class.

Geometry

• Classification of Euclidean plane isometries
• Classification theorem of surfaces
• * Classification of two-dimensional closed manifolds
• * Enriques–Kodaira classification of algebraic surfaces
• * Nielsen–Thurston classification which characterizes homeomorphisms of a compact surface
• Thurston's eight model geometries, and the geometrization conjecture

  Algebra

• Classification of finite simple groups
• Artin–Wedderburn theorem — a classification theorem for semisimple rings

  Linear algebra

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

  Complex analysis

• Classification of Fatou components