  This work and its companion paper, Amon et al. (2021), present cosmic shear
measurements and cosmological constraints from over 100 million source galaxies
in the Dark Energy Survey (DES) Year 3 data. We constrain the lensing amplitude
parameter $S_8\equiv\sigma_8\sqrt{\Omega_\textrm{m}/0.3}$ at the 3% level in
$\Lambda$CDM: $S_8=0.759^{+0.025}_{-0.023}$ (68% CL). Our constraint is at the
2% level when using angular scale cuts that are optimized for the $\Lambda$CDM
analysis: $S_8=0.772^{+0.018}_{-0.017}$ (68% CL). With cosmic shear alone, we
find no statistically significant constraint on the dark energy
equation-of-state parameter at our present statistical power. We carry out our
analysis blind, and compare our measurement with constraints from two other
contemporary weak-lensing experiments: the Kilo-Degree Survey (KiDS) and
Hyper-Suprime Camera Subaru Strategic Program (HSC). We additionally quantify
the agreement between our data and external constraints from the Cosmic
Microwave Background (CMB). Our DES Y3 result under the assumption of
$\Lambda$CDM is found to be in statistical agreement with Planck 2018, although
favors a lower $S_8$ than the CMB-inferred value by $2.3\sigma$ (a $p$-value of
0.02). This paper explores the robustness of these cosmic shear results to
modeling of intrinsic alignments, the matter power spectrum and baryonic
physics. We additionally explore the statistical preference of our data for
intrinsic alignment models of different complexity. The fiducial cosmic shear
model is tested using synthetic data, and we report no biases greater than
0.3$\sigma$ in the plane of $S_8\times\Omega_\textrm{m}$ caused by
uncertainties in the theoretical models.
