Most recently, boron-graphdiyne, a π-conjugated two-dimensional (2D) structure made from merely sp carbon skeleton connected with boron atoms was successfully experimentally realized through a bottom-to-up synthetic strategy. Motivated by this exciting experimental advance, we conducted density functional theory (DFT) and classical molecular dynamics simulations to study the mechanical, thermal conductivity and stability, electronic and optical properties of single-layer Bgraphdiyne. We particularly analyzed the application of this novel 2D material as an anode for Li, Na, Mg and Ca ions storage. Uniaxial tensile simulation results reveal that B-graphdiyne owing to its porous structure and flexibility can yield superstretchability. The single-layer B-graphdiyne was found to exhibit semiconducting electronic character, with a narrow band-gap of 1.15 eV based on the HSE06 prediction. It was confirmed that the mechanical straining can be employed to further tune the optical absorbance and electronic band-gap of B-graphdiyne. Ab initio molecular dynamics results reveal that B-graphdiyne can withstand at high temperatures, like 2500 K. The thermal conductivity of suspended single-layer Bgraphdiyne was predicted to be very low, ~2.5 W/mK at the room temperature. Our first-principles results reveal the outstanding prospect of B-graphdiyne as an anode material with ultrahigh charge capacities of 808 mAh/g, 5174 mAh/g and 3557 mAh/g for Na, Ca and Li ions storage, respectively. The comprehensive insight provided by this investigation highlights the outstanding physics of B-graphdiyne