This paper presents the discovery of Weyl semimetal (WSM) phases in non-centrosymmetric transition metal monophosphides, including TaAs, TaP, NbAs, and NbP. Using first-principle calculations, the authors show that these materials host 12 pairs of Weyl points in the Brillouin zone (BZ) without inversion symmetry. In the absence of spin-orbit coupling (SOC), band inversions in mirror invariant planes lead to gapless nodal rings. However, strong SOC opens full gaps in the mirror planes, generating nonzero mirror Chern numbers and Weyl points off the mirror planes. The resulting surface state Fermi arc structures on both (001) and (100) surfaces exhibit interesting shapes, indicating potential for future experimental studies. The WSM phase is defined by the presence of Weyl points, which are singular points of Berry curvature or "magnetic monopoles" in momentum space. These points appear when the spin doublet degeneracy of the bands is removed by breaking time reversal (T) or spatial inversion (P) symmetry. According to the "no-go Theorem," Weyl points always appear in pairs of opposite chirality or monopole charge. The conservation of chirality ensures the topological stability of WSM against perturbations that preserve translational symmetry. The authors propose that TaAs, TaP, NbAs, and NbP are natural WSMs with 12 pairs of Weyl points each. These materials are nonmagnetic, making them easier to study experimentally using ARPES. The Weyl points are determined by analyzing the mirror Chern numbers (MCN) and Z2 indices for the four mirror and time reversal invariant planes in the BZ. The WSM phase in this family is induced by a type of band inversion, which leads to nodal rings in the mirror plane. Once SOC is turned on, each nodal ring is gapped, except for three pairs of Weyl points, leading to fascinating physical properties, including complex Fermi arc structures on the surfaces. The surface states of these materials form complicated patterns for the Fermi surfaces, determined by the chirality distribution of the Weyl points and the MCNs for the mirror invariant planes. The authors also show that the surface states on the (001) and (100) surfaces exhibit unique Fermi arc structures, with the (001) surface having extremely long Fermi arcs that cross the zone boundary along the X to M line. These properties make the TaAs family of materials promising candidates for experimental study.This paper presents the discovery of Weyl semimetal (WSM) phases in non-centrosymmetric transition metal monophosphides, including TaAs, TaP, NbAs, and NbP. Using first-principle calculations, the authors show that these materials host 12 pairs of Weyl points in the Brillouin zone (BZ) without inversion symmetry. In the absence of spin-orbit coupling (SOC), band inversions in mirror invariant planes lead to gapless nodal rings. However, strong SOC opens full gaps in the mirror planes, generating nonzero mirror Chern numbers and Weyl points off the mirror planes. The resulting surface state Fermi arc structures on both (001) and (100) surfaces exhibit interesting shapes, indicating potential for future experimental studies. The WSM phase is defined by the presence of Weyl points, which are singular points of Berry curvature or "magnetic monopoles" in momentum space. These points appear when the spin doublet degeneracy of the bands is removed by breaking time reversal (T) or spatial inversion (P) symmetry. According to the "no-go Theorem," Weyl points always appear in pairs of opposite chirality or monopole charge. The conservation of chirality ensures the topological stability of WSM against perturbations that preserve translational symmetry. The authors propose that TaAs, TaP, NbAs, and NbP are natural WSMs with 12 pairs of Weyl points each. These materials are nonmagnetic, making them easier to study experimentally using ARPES. The Weyl points are determined by analyzing the mirror Chern numbers (MCN) and Z2 indices for the four mirror and time reversal invariant planes in the BZ. The WSM phase in this family is induced by a type of band inversion, which leads to nodal rings in the mirror plane. Once SOC is turned on, each nodal ring is gapped, except for three pairs of Weyl points, leading to fascinating physical properties, including complex Fermi arc structures on the surfaces. The surface states of these materials form complicated patterns for the Fermi surfaces, determined by the chirality distribution of the Weyl points and the MCNs for the mirror invariant planes. The authors also show that the surface states on the (001) and (100) surfaces exhibit unique Fermi arc structures, with the (001) surface having extremely long Fermi arcs that cross the zone boundary along the X to M line. These properties make the TaAs family of materials promising candidates for experimental study.