The ANAIS (Annual modulation with NaI(Tl) Scintillators) experiment aims at the confirmation or refutation of the DAMA/LIBRA positive annual modulation signal in the low energy detection rate, using the same target and technique, at the Canfranc Underground Laboratory (LSC) in Spain. ANAIS-112, consisting of nine 12.5 kg NaI(Tl) modules produced by Alpha Spectra Inc. in a 3 × 3 matrix configuration, is taking data smoothly in “dark matter search” mode since August, 2017, after a commissioning phase and operation of the first detectors during the last years in various set-ups. A large effort has been carried out within ANAIS to characterize the background of sodium iodide detectors, before unblinding the data and performing the first annual modulation analysis. Here, the background models developed for all the nine ANAIS-112 detectors are presented. Measured spectra from threshold to high energy in different conditions are well described by the models based on quantified activities independently estimated following several approaches. In the region from 1 to 6 keV ee the measured, efficiency corrected background level is 3.58 ± 0.02 keV - 1 kg - 1 day - 1 ; NaI crystal bulk contamination is the dominant background source being 210 Pb , 40 K , 22 Na and 3 H contributions the most relevant ones. This background level, added to the achieved 1 keV ee analysis threshold (thanks to the outstanding light collection and robust filtering procedures developed), allow ANAIS-112 to be sensitive to the modulation amplitude measured by DAMA/LIBRA, and able to explore at three sigma level in 5 years the WIMP parameter region singled out by this experiment.