The paper introduces XFOIL, an analysis and design system for low Reynolds number airfoils. It presents calculation procedures for viscous/inviscid analysis and mixed-inverse design of subcritical airfoils. The inviscid linear-vorticity panel method with a Karman-Tsien compressibility correction is developed for direct and mixed-inverse modes. Source distributions superimposed on the airfoil and wake allow modeling of the viscous layer influence on the potential flow. A two-equation lagged dissipation integral method is used to represent both laminar and turbulent viscous layers, with an $e^{\delta}$-type amplification formulation determining the transition point. The boundary layer and transition equations are solved simultaneously with the inviscid flowfield using a global Newton method. The system is particularly suitable for rapid analysis of low Reynolds number airfoil flows with transitional separation bubbles. The paper also outlines the basic inviscid and viscous formulations of XFOIL, demonstrates its performance on various airfoil cases, and describes the mixed-inverse formulation and associated user interface. The goal is to improve the productivity of the designer by reducing computational requirements while retaining the ability to predict low Reynolds number flows.The paper introduces XFOIL, an analysis and design system for low Reynolds number airfoils. It presents calculation procedures for viscous/inviscid analysis and mixed-inverse design of subcritical airfoils. The inviscid linear-vorticity panel method with a Karman-Tsien compressibility correction is developed for direct and mixed-inverse modes. Source distributions superimposed on the airfoil and wake allow modeling of the viscous layer influence on the potential flow. A two-equation lagged dissipation integral method is used to represent both laminar and turbulent viscous layers, with an $e^{\delta}$-type amplification formulation determining the transition point. The boundary layer and transition equations are solved simultaneously with the inviscid flowfield using a global Newton method. The system is particularly suitable for rapid analysis of low Reynolds number airfoil flows with transitional separation bubbles. The paper also outlines the basic inviscid and viscous formulations of XFOIL, demonstrates its performance on various airfoil cases, and describes the mixed-inverse formulation and associated user interface. The goal is to improve the productivity of the designer by reducing computational requirements while retaining the ability to predict low Reynolds number flows.