  Graded bundles are a particularly nice class of graded manifolds and
represent a natural generalisation of vector bundles. By exploiting the
formalism of supermanifolds to describe Lie algebroids we define the notion of
a weighted $A$-connection on a graded bundle. In a natural sense weighted
$A$-connections are adapted to the basic geometric structure of a graded bundle
in the same way as linear $A$-connections are adapted to the structure of a
vector bundle. This notion generalises directly to multi-graded bundles and in
particular we present the notion of a bi-weighted $A$-connection on a double
vector bundle. We prove the existence of such adapted connections and use them
to define (quasi-)actions of Lie algebroids on graded bundles.
