There are a number of basic definitions of K-theory: two coming from topology and two from algebraic geometry.
Given a compact Hausdorff space consider the set of isomorphism classes of finite-dimensional vector bundles over , denoted and let the isomorphism class of a vector bundle be denoted . Since isomorphism classes of vector bundles behave well with respect to direct sums, we can write these operations on isomorphism classes by

It should be clear that is an abelian monoid where the unit is given by the trivial vector bundle . We can then apply the Grothendieck completion to get an abelian group from this abelian monoid. This is called the K-theory of and is denoted .

We can use the Serre–Swan theorem and some algebra to get an alternative description of vector bundles over the ring of continuous complex-valued functions as projective modules. Then, these can be identified with idempotent matrices in some ring of matrices . We can define equivalence classes of idempotent matrices and form an abelian monoid . Its Grothendieck completion is also called . One of the main techniques for computing the Grothendieck group for topological spaces comes from the Atiyah–Hirzebruch spectral sequence, which makes it very accessible. The only required computations for understanding the spectral sequences are computing the group for the spheres .
There is an analogous construction by considering vector bundles in algebraic geometry. For a Noetherian scheme there is a set of all isomorphism classes of algebraic vector bundles on . Then, as before, the direct sum of isomorphisms classes of vector bundles is well-defined, giving an abelian monoid . Then, the Grothendieck group is defined by the application of the Grothendieck construction on this abelian monoid.
In algebraic geometry, the same construction can be applied to algebraic vector bundles over a smooth scheme. But, there is an alternative construction for any Noetherian scheme . If we look at the isomorphism classes of coherent sheaves we can mod out by the relation if there is a short exact sequence

This gives the Grothendieck-group which is isomorphic to if is smooth. The group is special because there is also a ring structure: we define it as

Using the Grothendieck–Riemann–Roch theorem, we have that

is an isomorphism of rings. Hence we can use for intersection theory.

The subject can be said to begin with Alexander Grothendieck (1957), who used it to formulate his Grothendieck–Riemann–Roch theorem. It takes its name from the German Klasse, meaning "class". Grothendieck needed to work with coherent sheaves on an algebraic variety X. Rather than working directly with the sheaves, he defined a group using isomorphism classes of sheaves as generators of the group, subject to a relation that identifies any extension of two sheaves with their sum. The resulting group is called K(X) when only locally free sheaves are used, or G(X) when all are coherent sheaves. Either of these two constructions is referred to as the Grothendieck group; K(X) has cohomological behavior and G(X) has homological behavior.

If X is a smooth variety, the two groups are the same. If it is a smooth affine variety, then all extensions of locally free sheaves split, so the group has an alternative definition.

In topology, by applying the same construction to vector bundles, Michael Atiyah and Friedrich Hirzebruch defined K(X) for a topological space X in 1959, and using the Bott periodicity theorem they made it the basis of an extraordinary cohomology theory. It played a major role in the second proof of the Atiyah–Singer index theorem (circa 1962). Furthermore, this approach led to a noncommutative K-theory for C*-algebras.

Already in 1955, Jean-Pierre Serre had used the analogy of vector bundles with projective modules to formulate Serre's conjecture, which states that every finitely generated projective module over a polynomial ring is free; this assertion is correct, but was not settled until 20 years later. (Swan's theorem is another aspect of this analogy.)

The other historical origin of algebraic K-theory was the work of J. H. C. Whitehead and others on what later became known as Whitehead torsion.

There followed a period in which there were various partial definitions of higher K-theory functors. Finally, two useful and equivalent definitions were given by Daniel Quillen using homotopy theory in 1969 and 1972. A variant was also given by Friedhelm Waldhausen in order to study the algebraic K-theory of spaces, which is related to the study of pseudo-isotopies. Much modern research on higher K-theory is related to algebraic geometry and the study of motivic cohomology.

The corresponding constructions involving an auxiliary quadratic form received the general name L-theory. It is a major tool of surgery theory.

In string theory, the K-theory classification of Ramond–Ramond field strengths and the charges of stable D-branes was first proposed in 1997.