  We deduce in this paper the sufficient conditions for weak convergence of
centered and normed deviation of the u-statistics with values in the space of
the real valued continuous function defined on some compact metric space.
  We obtain also a non-asymptotic and non-improvable up to multiplicative
constant moment and exponential tail estimates for distribution for the uniform
norm of centered and naturally normed deviation of u-statistics by means of its
martingale representation.
  Our results are formulated in a very popular and natural terms of metric
entropy in the distance (distances) generated by the introduced random
processes (fields).
