In the mathematical field of model theory, a theory is called stable if it satisfies certain combinatorial restrictions on its complexity. Stable theories are rooted in the proof of Morley's categoricity theorem and were extensively studied as part of Saharon Shelah's classification theory, which showed a dichotomy that either the models of a theory admit a nice classification or the models are too numerous to have any hope of a reasonable classification. A first step of this program was showing that if a theory is not stable then its models are too numerous to classify.

Stable theories were the predominant subject of pure model theory from the 1970s through the 1990s, so their study shaped modern model theory and there is a rich framework and set of tools to analyze them. A major direction in model theory is "neostability theory," which tries to generalize the concepts of stability theory to broader contexts, such as simple and NIP theories.

A common goal in model theory is to study a first-order theory by analyzing the complexity of the Boolean algebras of (parameter) definable sets in its models. One can equivalently analyze the complexity of the Stone duals of these Boolean algebras, which are type spaces. Stability restricts the complexity of these type spaces by restricting their cardinalities. Since types represent the possible behaviors of elements in a theory's models, restricting the number of types restricts the complexity of these models.

Stability theory has its roots in Michael Morley's 1965 proof of Łoś's conjecture on categorical theories. In this proof, the key notion was that of a totally transcendental theory, defined by restricting the topological complexity of the type spaces. However, Morley showed that (for countable theories) this topological restriction is equivalent to a cardinality restriction, a strong form of stability now called ${\displaystyle \omega }$-stability, and he made significant use of this equivalence. In the course of generalizing Morley's categoricity theorem to uncountable theories, Frederick Rowbottom generalized ${\displaystyle \omega }$-stability by introducing ${\displaystyle \kappa }$-stable theories for some cardinal ${\displaystyle \kappa }$, and finally Shelah introduced stable theories.

Stability theory was much further developed in the course of Shelah's classification theory program. The main goal of this program was to show a dichotomy that either the models of a first-order theory can be nicely classified up to isomorphism using a tree of cardinal-invariants (generalizing, for example, the classification of vector spaces over a fixed field by their dimension), or are so complicated that no reasonable classification is possible. Among the concrete results from this classification theory were theorems on the possible spectrum functions of a theory, counting the number of models of cardinality ${\displaystyle \kappa }$ as a function of ${\displaystyle \kappa }$. Shelah's approach was to identify a series of "dividing lines" for theories. A dividing line is a property of a theory such that both it and its negation have strong structural consequences; one should imply the models of the theory are chaotic, while the other should yield a positive structure theory. Stability was the first such dividing line in the classification theory program, and since its failure was shown to rule out any reasonable classification, all further work could assume the theory to be stable. Thus much of classification theory was concerned with analyzing stable theories and various subsets of stable theories given by further dividing lines, such as superstable theories.

One of the key features of stable theories developed by Shelah is that they admit a general notion of independence called non-forking independence, generalizing linear independence from vector spaces and algebraic independence from field theory. Although non-forking independence makes sense in arbitrary theories, and remains a key tool beyond stable theories, it has particularly good geometric and combinatorial properties in stable theories. As with linear independence, this allows the definition of independent sets and of local dimensions as the cardinalities of maximal instances of these independent sets, which are well-defined under additional hypotheses. These local dimensions then give rise to the cardinal-invariants classifying models up to isomorphism.

Let T be a complete first-order theory.

For a given infinite cardinal ${\displaystyle \kappa }$, T is ${\displaystyle \kappa }$-stable if for every set A of cardinality ${\displaystyle \kappa }$ in a model of T, the set S(A) of complete types over A also has cardinality ${\displaystyle \kappa }$. This is the smallest the cardinality of S(A) can be, while it can be as large as ${\displaystyle 2^{\kappa }}$. For the case ${\displaystyle \kappa =\aleph _{0}}$, it is common to say T is ${\displaystyle \omega }$-stable rather than ${\displaystyle \aleph _{0}}$-stable.