In mathematics, a fixed-point theorem is a theorem that a mathematical function has a fixed point. At that fixed point, the function's input and output are equal.

This concept is not one theorem itself; it is a way to describe many other theorems.

• Atiyah–Bott fixed-point theorem
• Banach fixed-point theorem
• Borel fixed-point theorem
• Browder fixed-point theorem
• Brouwer fixed-point theorem
• Caristi fixed-point theorem
• Diagonal lemma, also called the fixed-point lemma, for making logical sentences that describe themselves.
• Fixed-point lemma for normal functions
• Fixed-point property
• Injective metric space
• Kakutani fixed-point theorem
• Kleene fixpoint theorem
• Knaster–Tarski theorem
• Lefschetz fixed-point theorem
• Nielsen fixed-point theorem
• Poincaré–Birkhoff theorem proves that a function has two fixed points
• Ryll-Nardzewski fixed-point theorem
• Schauder fixed-point theorem
• Topological degree theory
• Tychonoff fixed-point theorem