One of the primary goals of exoplanetary science is to detect small, temperate planets passing (transiting) in front of bright and quiet host stars. This enables the characterization of planetary sizes, orbits, bulk compositions, atmospheres and formation histories. These studies are facilitated by small and cool M dwarf host stars. Here we report the Transiting Exoplanet Survey Satellite (TESS)1 discovery of three small planets transiting one of the nearest and brightest M dwarf hosts observed to date, TOI-270 (TIC 259377017, with K-magnitude 8.3, and 22.5 parsecs away from Earth). The M3V-type star is transited by the super-Earth-sized planet TOI-270 b (1.247-0.083+0.089?R⊕) and the sub-Neptune-sized planets TOI-270 c (2.42 ± 0.13 R⊕) and TOI-270 d (2.13 ± 0.12 R⊕). The planets orbit close to a mean-motion resonant chain, with periods (3.36 days, 5.66 days and 11.38 days, respectively) near ratios of small integers (5:3 and 2:1). TOI-270 is a prime target for future studies because (1) its near-resonance allows the detection of transit timing variations, enabling precise mass measurements and dynamical studies; (2) its brightness enables independent radial-velocity mass measurements; (3) the outer planets are ideal for atmospheric characterization via transmission spectroscopy; and (4) the quietness of the star enables future searches for habitable zone planets. Altogether, very few systems with small, temperate exoplanets are as suitable for such complementary and detailed characterization as TOI-270.