  We review some recent progress on the research of the periodic orbits of the
N-body problem,and propose a numerical scheme to determine the spatial
doubly-symmetric periodic orbits (SDSPs for short). Both comet- and lunar-type
SDSPs in the circular restricted three-body problem are computed, as well as
the Hill-type SDSPs in Hill's lunar problem. Doubly symmetries are exploited so
that the SDSPs can be computed efficiently. The monodromy matrix can be
calculated by the information of one fourth period. The periodicity conditions
are solved by Broyden's method with a line-search, and the algorithm is
reviewed. Some numerical examples show that the scheme is very efficient. For a
fixed period ratio and a given acute angle, there exist sixteen cases of
initial values. For the restricted three-body problem, the cases of "Copenhagen
problem" and the Sun-Jupiter-asteroid model are considered. New SDSPs are also
numerically found in Hill's lunar problem. Though the period ratio should be
small theoretically, some new periodic orbits are found when the ratio is not
too small, and most of the searched SDSPs are linearly stable.
