The performance of ellipsoidally shaped lead-glass capillaries for focusing the polychromatic synchrotron beam produced by a bending magnet of the DORIS positron storage ring (Hasylab, Hamburg, Germany) is discussed. The size, intensity and energy distribution of the focused beam produced by such capillaries are compared with those of beams generated by means of straight borosilicate capillaries, indicating that beam sizes ofca. 4 µm at the sample surface can be obtained with a total flux density that is ca. ten times higher than when a collimated beam is employed. Synchrotron radiation with energies up to 60 keV is focused, leaving the original energy distribution of the white synchrotron beam virtually unchanged. The analytical characteristics of the µ-XRF set-up at Beamline L of Hasylab, when equipped with a lead-glass capillary, were investigated by means of NIST SRMs and indicate that interference-free absolute/relative detection limits in the 1–10 fg/0.8–2 ppm range are achievable from 100 µm silicate-type samples for the elements from Mn (Z=25) to Gd (Z=64) using their Kα lines within 1000 s counting time. Elemental yields are situated in the 10–100 counts s^–1 per 100 mA per (µg cm^–2) range. As illustrations of the type of investigations these highly energetic, micrometre-sized beams make possible, the two-dimensional mapping of the distribution of REEs (rare earth elements) and other heavy elements in geological igneous rock samples and the three-dimensional non-destructive analysis of heavy metals (such as V, Fe, Ni and Mo) in individual fly-ash particles by means of fluorescence microtomography are briefly described.