  A survey of goodness-of-fit and symmetry tests based on the characterization
properties of distributions is presented. This approach became popular in
recent years. In most cases the test statistics are functionals of
$U$-empirical processes. The limiting distributions and large deviations of new
statistics under the null hypothesis are described. Their local Bahadur
efficiency for various parametric alternatives is calculated and compared with
each other as well as with diverse previously known tests. We also describe new
directions of possible research in this domain.
