  Koenig and Xi introduced {\em affine cellular algebras}. Kleshchev and
Loubert showed that an important class of {\em infinite dimensional} algebras,
the KLR algebras $R(\Gamma)$ of finite Lie type $\Gamma$, are (graded) affine
cellular; in fact, the corresponding affine cell ideals are idempotent. This
additional property is reminiscent of the properties of {\em quasihereditary
algebras} of Cline-Parshall-Scott in a {\em finite dimensional} situation. A
fundamental result of Cline-Parshall-Scott says that a finite dimensional
algebra $A$ is quasihereditary if and only if the category of finite
dimensional $A$-modules is a {\em highest weight category}. On the other hand,
S. Kato and Brundan-Kleshchev-McNamara proved that the category of {\em
finitely generated graded} $R(\Gamma)$-modules has many features reminiscent of
those of a highest weight category. The goal of this paper is to axiomatize and
study the notions of an {\em affine quasihereditary algebra} and an {\em affine
highest weight category}. In particular, we prove an affine analogue of the
Cline-Parshall-Scott Theorem. We also develop {\em stratified} versions of
these notions.
