The effect of airfoil design parameters, such as airfoil thickness and camber, are well understood in steady-state aerodynamics. But this knowledge cannot be readily applied to the flapping flight in insects and birds: flow visualizations and computational analyses of flapping flight have identified that in many cases, a leading-edge vortex (LEV) contributes substantially to the generation of aerodynamic force. In flapping flight, very high angles of attack and partly separated flow are common features. Therefore, it is expected that airfoil design parameters affect flapping wing aerodynamics differently. Existing studies have focused on force measurements, which do not provide sufficient insight into the dominant flow features. To analyse the influence of wing morphology in slow-speed bird flight, the time-resolved three-dimensional flow field around different flapping wing models in translational motion at a Reynolds number of $22\,000<\mathit{Re}<26\,000$ was studied. The effect of several Strouhal numbers ($0.2<\mathit{St}<0.4$), camber and thickness on the flow morphology and on the circulation was analysed. A strong LEV was found on all wing types at high $\mathit{St}$. The vortex is stronger on thin wings and enhances the total circulation. Airfoil camber decreases the strength of the LEV, but increases the total bound circulation at the same time, due to an increase of the ‘conventional’ bound circulation at the inner half of the wing. The results provide new insights into the influence of airfoil shape on the LEV and force generation at low $\mathit{Re}$. They contribute to a better understanding of the geometry of vertebrate wings, which seem to be optimized to benefit from LEVs in slow-speed flight.