  We introduce a new jet algorithm called XCone, for eXclusive Cone, which is
based on minimizing the event shape N-jettiness. Because N-jettiness partitions
every event into N jet regions and a beam region, XCone is an exclusive jet
algorithm that always returns a fixed number of jets. We use a new "conical
geometric" measure for which well-separated jets are bounded by circles of
radius R in the rapidity-azimuth plane, while overlapping jet regions
automatically form nearest-neighbor "clover jets". This avoids the split/merge
criteria needed in inclusive cone algorithms. A key feature of XCone is that it
smoothly transitions between the resolved regime where the N signal jets of
interest are well separated and the boosted regime where they overlap. The
returned value of N-jettiness also provides a quality criterion of how
N-jet-like the event looks. We also discuss the N-jettiness factorization
theorems that occur for various jet measures, which can be used to compute the
associated exclusive N-jet cross sections. In a companion paper, the physics
potential of XCone is demonstrated using the examples of dijet resonances,
Higgs decays to bottom quarks, and all-hadronic top pairs.
