  In this work, the notion of partial representation of a Hopf algebra is
introduced and its relationship with partial actions of Hopf algebras is
explored. Given a Hopf algebra $H$, one can associate it to a Hopf algebroid
$H_{par}$ which has the universal property that each partial representation of
$H$ can be factorized by an algebra morphism from $H_{par}$. We define also the
category of partial modules over a Hopf algebra $H$, which is the category of
modules over its associated Hopf algebroid $H_{par}$. The Hopf algebroid
structure of $H_{par}$ enables us to enhance the category of partial $H$
modules with a monoidal structure and such that the algebra objects in this
category are the usual partial actions. Some examples of categories of partial
$H$ modules are explored. In particular we can describe fully the category of
partially $\mathbb{Z}_2$-graded modules.
