This study investigates the aerodynamic trade-o s of a box-wing aircraft configuration using highfidelity aerodynamic optimization. A total of five optimization studies are conducted, where each study extends the previous one by progressively adding a combination of design variables and constraints. Examples of design variables include wing twist and sectional shape; examples of constraints include trim and stability requirements. In all cases the objective is to minimize inviscid drag at a prescribed lift and a Mach number of 0.78. Aerodynamic functionals are evaluated based on the discrete solution of the Euler equations, which are tightly coupled with an adjoint methodology incorporating a gradient-based optimizer. For each study an equivalent conventional tube-and-wing baseline is similarly optimized in order to enable direct comparisons. It is found that the transonic box-wing aircraft considered here, whose height-to-span ratio is about 0.2, produces up to 43% less induced drag than its conventional counterpart. The unique capability of the box wing to redistribute its optimal lift distribution contributes significantly to this benefit as it enables trim and other constraints to be satisfied with almost no performance degradation. The impact of nonlinear aerodynamics on the box wing is explored further through a series of subsonic optimization studies.