A branch of mathematics which brings together ideas from algebraic geometry, linear algebra, and number
theory. In general, there are two main types of 
-theory: topological and algebraic.
Topological 
-theory
is the "true" 
-theory
in the sense that it came first. Topological 
-theory has to do with vector
bundles over topological spaces. Elements
of a 
-theory are stable
equivalence classes of vector bundles over a
topological space. You can put a ring
structure on the collection of stably equivalent
bundles by defining addition through the Whitney
sum, and multiplication through the tensor
product of vector bundles. This defines "the
reduced real topological 
-theory
of a space."
"The reduced 
-theory
of a space" refers to the same construction, but instead of real vector bundles, complex vector bundles are used. Topological 
-theory is significant because it forms a generalized cohomology
theory, and it leads to a solution to the vector fields on spheres problem, as well
as to an understanding of the 
-homeomorphism
of homotopy theory.
Algebraic 
-theory
is somewhat more involved. Swan (1962) noticed that there is a correspondence between
the category of suitably nice topological
spaces (something like regular T2-spaces) and C-*-algebras. The idea is to associate to every space the C-*-algebra of
continuous maps from
that space to the reals.
A vector bundle over a space has sections, and these sections can be multiplied by continuous
functions to the reals. Under Swan's correspondence,
vector bundles correspond to modules over the C-*-algebra of continuous
functions, the modules being the modules of sections
of the vector bundle. This study of modules
over C-*-algebra is the starting point of algebraic
-theory.
The Quillen-Lichtenbaum conjecture connects algebraic 
-theory
to Étale cohomology.
