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Higher-order, non-Gaussian aspects of the large-scale structure carryvaluable information on structure formation and cosmology, which iscomplementary to second-order statistics. In this work we measure second- andthird-order weak-lensing aperture-mass moments from CFHTLenS and combine thosewith CMB anisotropy probes. The third moment is measured with a significance of$2\sigma$. The combined constraint on $\Sigma_8 = \sigma_8 (\Omega_{\rmm}/0.27)^\alpha$ is improved by 10%, in comparison to the second-order only,and the allowed ranges for $\Omega_{\rm m}$ and $\sigma_8$ are substantiallyreduced. Including general triangles of the lensing bispectrum yields tighterconstraints compared to probing mainly equilateral triangles. Second- andthird-order CFHTLenS lensing measurements improve Planck CMB constraints on$\Omega_{\rm m}$ and $\sigma_8$ by 26% for flat $\Lambda$CDM. For a model withfree curvature, the joint CFHTLenS-Planck result is $\Omega_{\rm m} = 0.28 \pm0.02$ (68% confidence), which is an improvement of 43% compared to Planckalone. We test how our results are potentially subject to three astrophysicalsources of contamination: source-lens clustering, the intrinsic alignment ofgalaxy shapes, and baryonic effects. We explore future limitations of thecosmological use of third-order weak lensing, such as the nonlinear model andthe Gaussianity of the likelihood function.

CFHTLenS: Cosmological constraints from a combination of cosmic shear two-point and three-point correlations