  We present first results from the third GRavitational lEnsing Accuracy
Testing (GREAT3) challenge, the third in a sequence of challenges for testing
methods of inferring weak gravitational lensing shear distortions from
simulated galaxy images. GREAT3 was divided into experiments to test three
specific questions, and included simulated space- and ground-based data with
constant or cosmologically-varying shear fields. The simplest (control)
experiment included parametric galaxies with a realistic distribution of
signal-to-noise, size, and ellipticity, and a complex point spread function
(PSF). The other experiments tested the additional impact of realistic galaxy
morphology, multiple exposure imaging, and the uncertainty about a
spatially-varying PSF; the last two questions will be explored in Paper II. The
24 participating teams competed to estimate lensing shears to within systematic
error tolerances for upcoming Stage-IV dark energy surveys, making 1525
submissions overall. GREAT3 saw considerable variety and innovation in the
types of methods applied. Several teams now meet or exceed the targets in many
of the tests conducted (to within the statistical errors). We conclude that the
presence of realistic galaxy morphology in simulations changes shear
calibration biases by $\sim 1$ per cent for a wide range of methods. Other
effects such as truncation biases due to finite galaxy postage stamps, and the
impact of galaxy type as measured by the S\'{e}rsic index, are quantified for
the first time. Our results generalize previous studies regarding sensitivities
to galaxy size and signal-to-noise, and to PSF properties such as seeing and
defocus. Almost all methods' results support the simple model in which additive
shear biases depend linearly on PSF ellipticity.
