Bromine monofluoride (BrF) and its cation (BrF + ) have attracted much scientific attention because of their potential significance in the stratospheric ozone depletion and the development of chemical laser. Despite that the structure and spectroscopic properties of the 1 3 Π 0+ (B 3 Π 0+ ) and 1 3 Π 1 states of BrF have been experimentally investigated in the literature, theoretical investigations of BrF and its cation are relatively sparse. In this paper, the low-lying electronic states for BrF/BrF + were studied by means of relativistic multireference configuration interaction method (including Davidson correction). The spin–orbit coupling effect was considered by the state-interacting method with the full Breit-Pauli Hamiltonian. For BrF, the potential energy curves (PECs) of 12 Λ–S states and 23 Ω states generated from the Λ–S states were calculated. The avoided crossing mechanism of Ω=0 + states were analyzed by the variations of dominant Λ–S composition for Ω states at several different internuclear distances. For BrF + , the PECs of the ground states (X 2 Π 3/2 and X 2 Π 1/2 ) were computed. The spectroscopic constants of the bound states of BrF/BrF + were determined, which are in good agreement with previously available experimental results. Finally, the ionization energies from the neutral ground state (X 1 Σ + ) to different ionic states (X 2 Π 3/2 , and X 2 Π 1/2 ) were obtained.