  We introduce the concept of a graded bundle which is a natural generalization
of the concept of a vector bundle and whose standard examples are higher
tangent bundles T^nQ playing a fundamental role in higher order Lagrangian
formalisms. Graded bundles are graded manifolds in the sense that we can choose
an atlas whose local coordinates are homogeneous functions of degrees
0,1,...,n. We prove that graded bundles have a convenient equivalent
description as homogeneity structures, i.e. manifolds with a smooth action of
the multiplicative monoid of non-negative reals. The main result states that
each homogeneity structure admits an atlas whose local coordinates are
homogeneous. Considering a natural compatibility condition of homogeneity
structures we formulate, in turn, the concept of a double (r-tuple, in general)
graded bundle - a broad generalization of the concept of a double (r-tuple)
vector bundle. Double graded bundles are proven to be locally trivial in the
sense that we can find local coordinates which are simultaneously homogeneous
with respect to both homogeneity structures.
