We present a tomographic cosmic shear analysis of the Kilo-Degree Survey (KiDS) combined with the VISTA Kilo-Degree Infrared Galaxy Survey (VIKING). This is the first time that a full optical to near-infrared data set has been used for a wide-field cosmological weak lensing experiment. This unprecedented data, spanning 450 deg^2^, allows us to significantly improve the estimation of photometric redshifts, such that we are able to include robustly higher-redshift sources for the lensing measurement, and - most importantly - to solidify our knowledge of the redshift distributions of the sources. Based on a flat LCDM model we find S_8_=0.737^+0.040^_-0.036_ in a blind analysis from cosmic shear alone. The tension between KiDS cosmic shear and the Planck Legacy CMB measurements remains in this systematically more robust analysis, with S_8_ differing by 2.3{sigma}. This result is insensitive to changes in the priors on nuisance parameters for intrinsic alignment, baryon feedback, and neutrino mass. KiDS shear measurements are calibrated with a new, more realistic set of image simulations and no significant B-modes are detected in the survey, indicating that systematic errors are under control. When calibrating our redshift distributions by assuming the 30-band COSMOS-2015 photometric redshifts are correct (following the Dark Energy Survey and the Hyper Suprime-Cam Survey), we find the tension with Planck is alleviated. The robust determination of source redshift distributions remains one of the most challenging aspects for future cosmic shear surveys. Data products from this analysis are available at http://kids.strw.leidenuniv.nl.