High-resolution images obtained in Hα with the new Swedish Solar Telescope at La Palma, Spain, have been used for studies of fine-scale threads in solar filaments. The widths of the thin threads are ≤0.3 arcsec. The fact that the width of the thinnest threads is comparable to the diffraction limit of the telescope of about 0.14 arcsec, at the wavelength of Hα, suggests that even thinner threads may exist. Assuming that the threads represent thin magnetic strings, we conclude that only a small fraction of these are filled with observable absorbing plasma, at a given time. The absorbing plasma is continuously flowing along the thread structures at velocities 15± 10 kms−1, which suggests that the flows must be field-aligned. In one case where a bundle of thin threads appears to be rooted in the nearby photosphere, we find that the individual threads connects with intergranular, dark lanes in the photosphere. We do not find signs of typical network fields at the ‘roots’ of the fine threads, as normally evidenced by bright points in associated G-band images. It is suggested that filament threads are rooted in relatively weak magnetic fields.