  In this paper we introduce a class of mathematical objects called
\emph{extensors} and develop some aspects of their theory with considerable
detail. We give special names to several particular but important cases of
extensors. The \emph{extension,} \emph{adjoint} and \emph{generalization}
operators are introduced and their properties studied. For the so-called
$(1,1)$-extensors we define the concept of \emph{determinant}, and their
properties are investigated. Some preliminary applications of the theory of
extensors are presented in order to show the power of the new concept in
action. An useful formula for the inversion of $(1,1)$-extensors is obtained.
