We report results of two studies performed by the Belle collaboration in search for a light dark matter candidate. A search for the invisible decay of the \Upsilon (1S) via the \Upsilon (3S) \to \pi ⁺\pi ^-\Upsilon (1S) transition yields no signal and an upper limit for the branching fraction at the 90% confidence level is determined to be  B (\Upsilon (1S) \to invisible)  < 2.5 \times 10^-3. No significant signal is also observed for the rare decays B \to h^(*) \nu \overline \nu , where h^(*) stands for a light meson and we set upper limits on the branching fractions at 90% confidence level. The limits on B⁰ \to K^*0 \nu \overline \nu  and B⁺ \to K⁺\nu \overline \nu  decays are more stringent than the previous constraints, while the first searches for B⁰ \to K⁰ \nu \overline \nu ,  \pi ⁰\nu \overline \nu , \rho ⁰\nu \overline \nu ,  \phi \nu \overline \nu  and B⁺ \to K^*+\nu \overline \nu , \rho ⁺\nu \overline \nu  are reported.

PACS numbers: 13.25.Gv, 95.30.Cq, 13.25.Hw, 14.40.Nd