A front vehicle-body structure for a vehicle, comprising: a front windshield; a cowl provided to support the front windshield in an inclined state where a rear side of the front windshield is located at a substantially higher level than a front side of the front windshield and forming a lower edge of an opening portion for arranging the front windshield; a bonnet provided in front of and above the cowl; a dash panel provided below the cowl and configured to extend downwardly from a lower portion of the cowl; and a gate-shaped reinforcing member, wherein the cowl comprises a cowl upper panel and a cowl lower panel, wherein said cowl further comprises a first face potion to which said front windshield adheres and an inclined second face portion which is configured to extend obliquely rearwardly-and-downwardly from said first face portion, and a front end of said first face portion is positioned on a rearward side, in a vehicle longitudinal direction, of a rear end of said bonnet, wherein the cowl has a closed-cross section portion configured to extend in a vehicle width direction which is formed by faces including said first face portion and said second face portion, wherein a bottom face portion which forms a bottom face of said closed-cross section portion of the cowl is configured to extend obliquely rearwardly-and-downwardly, wherein an upper end portion of said dash panel is joined to said bottom face portion of the closed-cross section portion of the cowl from below, wherein the dash panel comprises a dash-panel body portion configured to extend in a vehicle vertical direction in a substantially vertical-wall shape, wherein the gate-shaped reinforcing member comprises a vehicle-width-direction extension member which is configured to extend in the vehicle width direction and a pair of right-and-left vehicle-vertical-direction extension members which are configured to downwardly extend substantially vertically from right-and-left both ends of the vehicle-width-direction extension member, which is formed in a gate shape which is opened downwardly in the elevational view, and wherein the vehicle-width-direction extension member is provided below the cowl lower panel in an area located above the dash panel such that the vehicle-width-direction extension member extends, in the vehicle longitudinal direction, over the cowl which is provided to project forwardly beyond the dash-panel body portion. a front windshield; a cowl provided to support the front windshield in an inclined state where a rear side of the front windshield is located at a substantially higher level than a front side of the front windshield and forming a lower edge of an opening portion for arranging the front windshield; a bonnet provided in front of and above the cowl; a dash panel provided below the cowl and configured to extend downwardly from a lower portion of the cowl; and a gate-shaped reinforcing member, wherein the cowl comprises a cowl upper panel and a cowl lower panel, wherein said cowl further comprises a first face potion to which said front windshield adheres and an inclined second face portion which is configured to extend obliquely rearwardly-and-downwardly from said first face portion, and a front end of said first face portion is positioned on a rearward side, in a vehicle longitudinal direction, of a rear end of said bonnet, wherein the cowl has a closed-cross section portion configured to extend in a vehicle width direction which is formed by faces including said first face portion and said second face portion, wherein a bottom face portion which forms a bottom face of said closed-cross section portion of the cowl is configured to extend obliquely rearwardly-and-downwardly, wherein an upper end portion of said dash panel is joined to said bottom face portion of the closed-cross section portion of the cowl from below, wherein the dash panel comprises a dash-panel body portion configured to extend in a vehicle vertical direction in a substantially vertical-wall shape, wherein the gate-shaped reinforcing member comprises a vehicle-width-direction extension member which is configured to extend in the vehicle width direction and a pair of right-and-left vehicle-vertical-direction extension members which are configured to downwardly extend substantially vertically from right-and-left both ends of the vehicle-width-direction extension member, which is formed in a gate shape which is opened downwardly in the elevational view, and wherein the vehicle-width-direction extension member is provided below the cowl lower panel in an area located above the dash panel such that the vehicle-width-direction extension member extends, in the vehicle longitudinal direction, over the cowl which is provided to project forwardly beyond the dash-panel body portion.