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We examine the cosmological constraining power of future large-scale weaklensing surveys on the model of \emph{Euclid}, with particular reference toprimordial non-Gaussianity. Our analysis considers several different estimatorsof the projected matter power spectrum, based on both shear and flexion, forwhich we review the covariances and Fisher matrices. The bounds provided bycosmic shear alone for the local bispectrum shape, marginalized over$\sigma_8$, are at the level of $\Delta f_\mathrm{NL} \sim 100$. We considerthree additional bispectrum shapes, for which the cosmic shear constraintsrange from $\Delta f_\mathrm{NL}\sim 340$ (equilateral shape) up to $\Deltaf_\mathrm{NL}\sim 500$ (orthogonal shape). The competitiveness of cosmicflexion constraints against cosmic shear ones depends on the galaxy intrinsicflexion noise, that is still virtually unconstrained. Adopting the very highvalue that has been occasionally used in the literature results in the flexioncontribution being basically negligible with respect to the shear one, and forrealistic configurations the former does not improve significantly theconstraining power of the latter. Since the flexion noise decreases withdecreasing scale, by extending the analysis up to $\ell_\mathrm{max} = 20,000$cosmic flexion, while being still subdominant, improves the shear constraintsby $\sim 10%$ when added. However on such small scales the highly non-linearclustering of matter and the impact of baryonic physics make any errorestimation uncertain. By considering lower, and possibly more realistic, valuesof the flexion intrinsic shape noise results in flexion constraining powerbeing a factor of $\sim 2$ better than that of shear, and the bounds on$\sigma_8$ and $f_\mathrm{NL}$ being improved by a factor of $\sim 3$ upontheir combination. (abridged)

Constraining primordial non-Gaussianity with cosmological weak lensing: shear and flexion