We present here the discovery of a new class of superslow rotating asteroids (Prot ≳1000 h) in data extracted from the Asteroid Terrestrial-impact Last Alert System (ATLAS) and Zwicky Transient Facility (ZTF) all-sky surveys. Of the 39 rotation periods we report here, 32 have periods longer than any previously reported unambiguous rotation periods currently in the Asteroid Light Curve Data base. In our sample, seven objects have a rotation period >4000 h and the longest period we report here is 4812 h (~200 d). We do not observe any correlation between taxonomy, albedo, or orbital properties with superslow rotating status. The most plausible mechanism for the creation of these very slow rotators is if their rotations were slowed by YORP spin-down. Superslow rotating asteroids may be common, with at least 0.4 per cent of the main-belt asteroid population with a size range between 2 and 20 km in diameter rotating with periods longer than 1000 h.