  Curves of genus g which admit a map to CP1 with specified ramification
profile mu over 0 and nu over infinity define a double ramification cycle
DR_g(mu,nu) on the moduli space of curves. The study of the restrictions of
these cycles to the moduli of nonsingular curves is a classical topic. In 2003,
Hain calculated the cycles for curves of compact type. We study here double
ramification cycles on the moduli space of Deligne-Mumford stable curves.
  The cycle DR_g(mu,nu) for stable curves is defined via the virtual
fundamental class of the moduli of stable maps to rubber. Our main result is
the proof of an explicit formula for DR_g(mu,nu) in the tautological ring
conjectured by Pixton in 2014. The formula expresses the double ramification
cycle as a sum over stable graphs (corresponding to strata classes) with
summand equal to a product over markings and edges. The result answers a
question of Eliashberg from 2001 and specializes to Hain's formula in the
compact type case.
  When mu and nu are both empty, the formula for double ramification cycles
expresses the top Chern class lambda_g of the Hodge bundle of the moduli space
of stable genus g curves as a push-forward of tautological classes supported on
the divisor of nonseparating nodes. Applications to Hodge integral calculations
are given.
