  In this paper we study the mean-field backward stochastic differential
equations (mean-field bsde) of the form dY(t) =-f(t,Y(t),Z(t),K(t, .
),E[\varphi(Y(t),Z(t),K(t,.))])dt+Z(t)dB(t)
+\int_{R_{0}}K(t,\zeta)\tilde{N}(dt,d\zeta), where B is a Brownian motion,
\tilde{N} is the compensated Poisson random measure.
  Under some mild conditions, we prove the existence and uniqueness of the
solution triplet (Y,Z,K). It is commonly believed that there is no comparison
theorem for general mean-field bsde. However, we prove a comparison theorem for
a subclass of these equations. When the mean-field bsde is linear, we give an
explicit formula for the first component Y(t) of the solution triplet. Our
results are applied to solve a mean-field recursive utility optimization
problem in finance.
