An LED floodlight fixture having a housing and having its two major principal axes (54, 48) in a fixture plane the dimensions of the fixture in planes parallel to its third principal axis being substantially smaller than the largest dimensions parallel to the fixture plane, the fixture comprising: a first outer surface having a first central portion; an opposite second outer surface having a second central portion substantially aligned with the first central portion and encompassing a light-emitting region, characterized in that the light-emitting region includes an LED assembly having at least one LED-array module mounted on an LED heat sink, the LED assembly being secured with respect to the housing adjacent thereto in non-water/air-tight condition permitting air/water-flow to and from the LED assembly, the housing includes a perimetrical structure having first and second edge-adjacent portions (22, 28) of the first and second outer surfaces (18, 24), the first and second edge-adjacent portions (22, 28) each having a boundary (46, 52) with the respective one of the first and second central portions (20, 44), said boundary of the second edge-adjacent portion with the second central portion defining a reference plane, the first and second edge-adjacent portions (22, 28) are substantially convex and meet at the perimetrical edge to form aerodynamic-drag-reducing cross-sectional profiles transverse to the fixture plane and extending in substantially all fixture-plane directions from the intersection of the two major principal axes (54, 48), the first and second edge-adjacent portions (22, 28) together having ratios of the maximum dimension of the profile in a direction parallel to the third principal axis to the maximum dimension of the profile in the fixture plane of about 3 or less; each of the first and second central portions (20, 44) extend across at least 25% of an area within the perimetrical edge; and the greatest dimension between the first central portion and the reference plane is no more than 50% greater than the smallest dimension therebetween. a first outer surface having a first central portion; an opposite second outer surface having a second central portion substantially aligned with the first central portion and encompassing a light-emitting region, characterized in that the light-emitting region includes an LED assembly having at least one LED-array module mounted on an LED heat sink, the LED assembly being secured with respect to the housing adjacent thereto in non-water/air-tight condition permitting air/water-flow to and from the LED assembly, the housing includes a perimetrical structure having first and second edge-adjacent portions (22, 28) of the first and second outer surfaces (18, 24), the first and second edge-adjacent portions (22, 28) each having a boundary (46, 52) with the respective one of the first and second central portions (20, 44), said boundary of the second edge-adjacent portion with the second central portion defining a reference plane, the first and second edge-adjacent portions (22, 28) are substantially convex and meet at the perimetrical edge to form aerodynamic-drag-reducing cross-sectional profiles transverse to the fixture plane and extending in substantially all fixture-plane directions from the intersection of the two major principal axes (54, 48), the first and second edge-adjacent portions (22, 28) together having ratios of the maximum dimension of the profile in a direction parallel to the third principal axis to the maximum dimension of the profile in the fixture plane of about 3 or less; each of the first and second central portions (20, 44) extend across at least 25% of an area within the perimetrical edge; and the greatest dimension between the first central portion and the reference plane is no more than 50% greater than the smallest dimension therebetween.