  We present cosmological parameter constraints from a tomographic weak
gravitational lensing analysis of ~450deg$^2$ of imaging data from the Kilo
Degree Survey (KiDS). For a flat $\Lambda$CDM cosmology with a prior on $H_0$
that encompasses the most recent direct measurements, we find
$S_8\equiv\sigma_8\sqrt{\Omega_{\rm m}/0.3}=0.745\pm0.039$. This result is in
good agreement with other low redshift probes of large scale structure,
including recent cosmic shear results, along with pre-Planck cosmic microwave
background constraints. A $2.3$-$\sigma$ tension in $S_8$ and `substantial
discordance' in the full parameter space is found with respect to the Planck
2015 results. We use shear measurements for nearly 15 million galaxies,
determined with a new improved `self-calibrating' version of $lens$fit
validated using an extensive suite of image simulations. Four-band $ugri$
photometric redshifts are calibrated directly with deep spectroscopic surveys.
The redshift calibration is confirmed using two independent techniques based on
angular cross-correlations and the properties of the photometric redshift
probability distributions. Our covariance matrix is determined using an
analytical approach, verified numerically with large mock galaxy catalogues. We
account for uncertainties in the modelling of intrinsic galaxy alignments and
the impact of baryon feedback on the shape of the non-linear matter power
spectrum, in addition to the small residual uncertainties in the shear and
redshift calibration. The cosmology analysis was performed blind. Our
high-level data products, including shear correlation functions, covariance
matrices, redshift distributions, and Monte Carlo Markov Chains are available
at http://kids.strw.leidenuniv.nl.
