  For 0-cycles on a variety over a number field, we define an analogue of the
classical descent set for rational points. This leads to, among other things, a
definition of the \'etale-Brauer obstruction set for 0-cycles, which we show is
contained in the Brauer-Manin set and is compatible with Suslin's singular
homology of degree 0. We then transfer some tools and techniques used to study
the arithmetic of rational points into the setting of 0-cycles. For example, we
extend the strategy developed by Y. Liang, relating the arithmetic of rational
points over finite extensions of the base field to that of 0-cycles, to
torsors. We give applications of our results to study the arithmetic behaviour
of 0-cycles for Enriques surfaces, torsors given by (twisted) Kummer varieties,
universal torsors, and torsors under tori.
