to produce $S=-2$ systems (E176 and E373 at KEK, E885 and E906 at BNL) have been performed and some of the data analyses are still in progress to get novel information on the $S=-2$ interactions.

In this exciting situation of the experimental study, it is needed to perform careful theoretical calculations of double-$\Lambda$ hypernuclei with refreshed viewpoints. As one of the motivations of the present work, we think it necessary and timely to put the NAGARA data of $_\Lambda^{ }$$_\Lambda^6$He binding energy as a new standard basis for a systematic study of a series of several double-$\Lambda$ species. Secondly, in order to extract information on the $\Lambda \Lambda$ interactions precisely, here we emphasize that hypernuclear calculations should be complete and realistic enough to leave structural ambiguity as negligibly as possible. All the dynamical changes due to successive $\Lambda$ participation should be also taken faithfully. To meet these requirement we explore light $p$-shell double-$\Lambda$ hypernuclei ($A=6-10)$ comprehensively using the microscopic three- and four-body models. Thirdly, by these systematic and realistic calculations, we will give reliable prediction of not only the ground-state binding energies but also possible excited-state energies, which encourages double-$\Lambda$ hypernuclear spectroscopic study in near future.

So far several cluster 