Most three-dimensional (3D) and two-dimensional (2D) boron nitride (BN) structures are wide-band-gap insulators.Here, we propose two BN monolayers having Dirac points and fl at bands, respectively. One monolayer is named as 5-7 BN that consists of five-and seven-membered rings. The other is a Kagome BN made of triangular boron rings and nitrogen dimers. The two structures show not only good dynamic and thermodynamic stabilities but also novel electronic properties. The 5-7 BN has Dirac points on the Fermi level, indicating that the structure is a typical Dirac material. The Kagome BN has double fl at bands just below the Fermi level, and thus there are heavy fermions in the structure. The fl at-band-induced ferromagnetism is also revealed. We analyze the origination of the band structures by partial density of states and projection of orbitals. In addition, a possible route to experimentally grow the two structures on some suitable substrates such as the PbO2 (111) surface and the CdO (111) surface is also discussed, respectively. Our research not only extends understanding on the electronic properties of BN structures, but also may expand the applications of BN materials in 2D electronic devices.