Universal algebra (sometimes called general algebra) is the field of mathematics that studies algebraic structures in general, not specific types of algebraic structures. For instance, rather than considering groups or rings as the object of study—this is the subject of group theory and ring theory— in universal algebra, the object of study is the possible types of algebraic structures and their relationships.

In universal algebra, an algebra (or algebraic structure) is a set A together with a collection of operations on A.

An n-ary operation on A is a function that takes n elements of A and returns a single element of A. Thus, a 0-ary operation (or nullary operation) can be represented simply as an element of A, or a constant, often denoted by a letter like a. A 1-ary operation (or unary operation) is simply a function from A to A, often denoted by a symbol placed in front of its argument, like ~x. A 2-ary operation (or binary operation) is often denoted by a symbol placed between its arguments (also called infix notation), like x ∗ y. Operations of higher or unspecified arity are usually denoted by function symbols, with the arguments placed in parentheses and separated by commas, like f(x,y,z) or f(x₁,...,x_n). One way of talking about an algebra, then, is by referring to it as an algebra of a certain type ${\displaystyle \Omega }$, where ${\displaystyle \Omega }$ is an ordered sequence of natural numbers representing the arity of the operations of the algebra. However, some researchers also allow infinitary operations, such as ${\displaystyle \textstyle \bigwedge _{\alpha \in J}x_{\alpha }}$ where J is an infinite index set, which is an operation in the algebraic theory of complete lattices.

After the operations have been specified, the nature of the algebra is further defined by axioms, which in universal algebra often take the form of identities, or equational laws. An example is the associative axiom for a binary operation, which is given by the equation x ∗ (y ∗ z) = (x ∗ y) ∗ z. The axiom is intended to hold for all elements x, y, and z of the set A.

A collection of algebraic structures defined by identities is called a variety or equational class.

Restricting one's study to varieties rules out:

The study of equational classes can be seen as a special branch of model theory, typically dealing with structures having operations only (i.e. the type can have symbols for functions but not for relations other than equality), and in which the language used to talk about these structures uses equations only.

Not all algebraic structures in a wider sense fall into this scope. For example, ordered groups involve an ordering relation, so would not fall within this scope.