Using a three-body model with density-dependent contact interaction, we discuss the role of di-neutron correlations in the ground state properties as well as in the dipole excitations of typical weakly-bound Borromean nuclei, $^{11}$Li, and $^6$He.We show that, while both the nuclei manifest themselves similar strong di-neutron correlations to each other in the ground state, the energy distributions for the two emitted neutrons from
the dipole excitation are considerably different. We also discuss the di-proton correlation in a proton-rich Borromean nucleus, $^{17}$Ne. Possible BCS-BEC crossover phenomena in nuclear matter and borromian nuclei will be pointed out.