Dark matter dominates the mass distribution of the universe, and dark energy determines its expansion. The two are the most mysterious and attractive subjects in modern cosmology, because they provide an opportunity to discover new fundamental physics.

Cosmological weak gravitational lensing, which describes the deflection of photons by the gravitational force from large-scale structure in the universe, has been an active area of research in the past decade with many completed, ongoing, and upcoming surveys. Because weak lensing is sensitive to the growth of structure and expansion history of the universe, it is a great tool for improving our understanding of both dark matter and dark energy problems.

Cosmic structures have become non-linear by gravitational clustering. The non-linear structures are important to weak lensing, and cause non-Gaussianity in the lensing maps. In this thesis, I study the influence of non-linearity and non-Gaussianity on the uncertainty of lensing measurements. I develop a new method to robustly measure the covariance matrix of the lensing convergence power spectrum, from simulations. Because 21-cm intensity map may soon cover half sky at redshift 1-4, I build optimal estimators for reconstructing lensing from the 21-cm sources. I develop Gaussian optimal estimators which can be derived analytically, and non-Gaussian optimal estimators which can be constructed numerically from simulation data. I then run a large number of N-body simulations. For both lenses and 21-cm sources, I explore the statistical uncertainties in ii the simulation data.

We show that the non-Gaussianity nature of lensing decreases the dark energy figure of merit by a factor of 1.3 to 1.6 for a few future surveys. We also find that the non-Gaussianity nature of the 21-cm sources reduces the signal to noise ratio by several orders of magnitude. The reconstruction noise saturates at mildly non-linear scales, where the linear power spectrum of the source is Delta^2 ~ 0.2 - 0.5. For 21-cm sources at z ~ 2-4, the lensing reconstruction is limited by cosmic variance at ell <~ 100, which is in the linear regime of gravitational growth, and robustly predicted by theory. This allows promising constraints to various modified gravity and dark energy models.