We perform wavefront measurements of high-order harmonics using an extreme-ultraviolet (XUV) Hartmann sensor and study how their spatial properties vary with different generation parameters, such as pressure in the nonlinear medium, fundamental pulse energy and duration as well as beam size. In some conditions, excellent wavefront quality (up to λ/11) was obtained. The high throughput of the intense XUV beamline at the Lund Laser Centre allows us to perform single-shot measurements of both the full harmonic beam generated in argon and individual harmonics selected by multilayer mirrors. We theoretically analyze the relationship between the spatial properties of the fundamental and those of the generated high-order harmonics, thus gaining insight into the fundamental mechanisms involved in high-order harmonic generation (HHG).