  In this paper we introduce and study a new kind of hyperbolic geometric flows
--dissipative hyperbolic geometric flow. This kind of flow is defined by a
system of quasilinear wave equations with dissipative terms. Some interesting
exact solutions are given, in particular, a new concept-- hyperbolic Ricci
soliton is introduced and some of its geometric properties are described. We
also establish the short-time existence and uniqueness theorem for the
dissipative hyperbolic geometric flow, and prove the nonlinear stability of the
flow defined on the Euclidean space of dimension larger than 2. Wave character
of the evolving metrics and curvatures is illustrated and the nonlinear wave
equations satisfied by the curvatures are derived.
