  We develop a systematic approach to contact and Jacobi structures on graded
supermanifolds. In this framework, contact structures are interpreted as
symplectic principal GL(1,R)-bundles. Gradings compatible with the
GL(1,R)-action lead to the concept of a graded contact manifold, in particular
a linear (more generally, n-linear) contact structure. Linear contact
structures are proven to be exactly the canonical contact structures on first
jets of line bundles. They provide linear Kirillov (or Jacobi) brackets and
give rise to the concept of a Kirillov algebroid, an analog of a Lie algebroid,
for which the corresponding cohomology operator is represented not by a vector
field (de Rham derivative) but a first-order differential operator. It is shown
that one can view Kirillov or Jacobi brackets as homological Hamiltonians on
linear contact manifolds. Contact manifolds of degree 2 are studied, as well as
contact analogs of Courant algebroids. We define lifting procedures that
provide us with constructions of canonical examples of the structures in
question.
