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We construct a comprehensive model atom for TiI-II using more than 3600measured and predicted energy levels of TiI and 1800 energy levels of TiII, andquantum mechanical photoionisation cross-sections. Non-local thermodynamicalequilibrium (NLTE) line formation for the two ions of titanium is treatedthrough a wide range of spectral types from A to K, including metal-poor starswith [Fe/H] down to -2.6 dex. NLTE leads to systematically depleted totalabsorption in the TiI lines and to positive abundance corrections. Themagnitude of NLTE abundance corrections is smaller compared to the availableliterature data for the cool (FGK type) atmospheres. NLTE leads to strengthenedTiII lines and to negative NLTE abundance corrections. For the first time, weperformed the NLTE calculations for TiI-II in the 6500 K < Teff < 13000 Krange. The deviations from LTE grow towards higher effective temperature. Forfour A type stars we derived the NLTE and LTE abundances from lines of TiI andTiII. In LTE an abundance discrepancy of up to 0.22 dex was obtained betweenTiI and TiII, and it vanishes in NLTE. For other four A-B stars, with only TiIIlines observed, NLTE leads to smaller line-to-line scatter compared to LTE. Anefficiency of inelastic TiI + HI collisions was empirically estimated fromanalysis of TiI and TiII lines in 17 cool stars in a wide metallicity range-2.6 < [Fe/H] < 0.0. Consistent NLTE abundances from lines of TiI and TiII wereobtained applying classical Drawinian rates for the stars with log g > 4.1, andneglecting inelastic collisions with HI for the VMP giant HD 122563. For thevery metal-poor turn-off stars ([Fe/H] < -2 and log g < 4.1), the abundancedifference TiI-TiII was obtained to be positive already in LTE and it increasesin NLTE. The accurate collisional data for TiI and TiII are desired to find aclue to this problem.

A NLTE line formation for neutral and singly-ionised titanium in model atmospheres of the reference A-K stars