  We state results from noncommutative deformation theory of modules over an
associative $k$-algebra $A,$ $k$ a field, necessary for this work. We define a
set of $A$-modules $\operatorname{aSpec}A$ containing the simple modules, whose
elements we call spectral, for which there exists a topology where the simple
modules are the closed points. Applying results from deformation theory we
prove that there exists a sheaf of associative rings $\mathcal O_X$ on the
topological space $X=\operatorname{aSpec}A$ giving it the structure of a
pointed ringed space. In general, an associative variety $X$ is a ringed space
with an open covering $\{U_i=\operatorname{aSpec}{A_i}\}_{i\in I}.$ When $A$ is
a commutative $k$-algebra, $\operatorname{aSpec}A\simeq\spec A,$ and so the
category $\cat{aVar}_k$ of associative varieties is an extension of the
category of varieties $\cat{Var}_k,$ i.e. there exists a faithfully full
functor $I:\cat{Var}_k\rightarrow\cat{aVar}_k.$ Our main result says that any
associative variety $X$ is $\operatorname{aSpec}(\mathcal O_X(X))$ for the
$k$-algebra $\mathcal O_X(X),$ and so any study of varieties can be reduced to
the study of the associative algebra $\mathcal O_X(X).$
