  We explore the dependence of weak lensing phenomena on the background
cosmology. We first generalise the relation between $P_\psi(\omega)$, the
angular power spectrum of the distortion, and the power spectrum of density
fluctuations to non-flat cosmologies. We then compute $P_\psi$ for various
illustrative models. A useful cosmological discriminator is the growth of
$P_\psi$ with source redshift which is much stronger in low matter density
models, and especially in $\Lambda$-dominated models. With even crude redshift
information (say from broad band colours) it should be possible to constrain
the cosmological world model. The amplitude of $P_\psi(\omega)$ is also quite
sensitive to the cosmology, but requires a reliable external normalisation for
the mass fluctuations. If one normalises to galaxy clustering, with $M/L$ fixed
by small-scale galaxy dynamics, then low density models predict a much stronger
distortion. If, however, one normalises to large-scale bulk-flows, the
predicted distortion for sources at redshifts $Z_s \sim 1-3$ is rather
insensitive to the background cosmology. The signals predicted here can be
detected at a very high level of significance with a photometric survey
covering say 10 square degrees, but sparse sampling is needed to avoid large
sampling variance and we discuss the factors influencing the design of an
optimum survey. Turning to weak lensing by clusters we find that for high lens
redshifts ($Z_l\simeq1$) the critical density is substantially reduced in
$\Lambda$ models, but that the ratio of the shear or convergence to the
velocity dispersions or X-ray temperature of clusters is only very weakly
dependent on the cosmology.

Astrophysics