As an important two-dimensional material which shows exceptional mechanical and chemical stability, superior electronic properties, along with broad applications, the hexagonal-BN (h-BN) has drawn great attention recently. Here we report a systematic study on the structural stability, electronic and magnetic properties of various h-BN edges, including both bare and hydrogen-terminated ones. It is found that along the armchair (AC) direction, the pristine edge is the most stable one because of the formation of a triple B≡N bond, while, along the zigzag (ZZ) directions, the reconstructed ones, ZZB + N and ZZN57 are more stable. The pristine edges are more stable in bare BN in most cases if saturated with hydrogen. By applying the theory of Wulff construction, we predicted that an unpassivated BN domain prefers the hexagonal shape enclosed with bare AC edges i.e., AC-Ns, AC, AC-Bs if the feedstock varies from N-rich to B-rich. However, the evolution from ZZN edged triangular domain, to hexagonal domain enclosed with AC edges, and ZZB edged triangle may occur if the edges are terminated by hydrogen atoms. Further calculation shows that these edges present rich type-dependent properties and thus are important for various applications. This theoretical study showed that controlling the morphologies of BN domains and BN edges is crucial for various applications.