We have performed systematic high-resolution angle-resolved photoemission spectroscopy (ARPES) on two novel boride compounds, MgB2 and CaB6, to study the electronic structure near the Fermi level relevant to the anomalous physical properties (superconductivity and ferromagnetism) observed in these compounds. In MgB2, we have succeeded in separately observing two superconducting gaps which originate in the σ and π bands, respectively, and thereby established the two-band superconductivity. ARPES measurements on AlB2, which is a non-superconducting isostructural reference compound of MgB2 with an additional electron in a unit cell, indicates the importance of the σ band for the superconductivity. In CaB6, we observed an energy gap of 1 eV at the X point in the Brillouin zone, which suggests that CaB6 is intrinsically a large-gap semiconductor. This observation requires reconsideration of the excitonic-insulator model to explain the anomalous ferromagnetism in CaB6.