  Weak gravitational lensing (WL) causes distortions of galaxy images and
probes massive structures on large scales, allowing us to understand the
late-time evolution of the Universe. One way to extract the cosmological
information from WL is to use peak statistics. Peaks are tracers of massive
halos and therefore probe the mass function. They retain non-Gaussian
information and have already been shown as a promising tool to constrain
cosmology. In this work, we develop a new model to predict WL peak counts. The
model generates fast simulations based on halo sampling and selects peaks from
the derived lensing maps. This approach has three main advantages. First, the
model is very fast: only several seconds are required to perform a realization.
Second, including realistic conditions is straightforward. Third, the model
provides the full distribution information because of its stochasticity. We
show that our model agrees well with N-body simulations. Then, we study the
impacts of the cosmology-dependent covariance on constraints and explore
different parameter inference methods. A special focus is put on approximate
Bayesian computation (ABC), an accept-reject sampler without the need to
estimate the likelihood. We show that ABC is able to yield robust constraints
with much reduced time costs. Several filtering techniques are studied to
improve the extraction of multiscale information. Finally, the new model is
applied to the CFHTLenS, KiDS DR1/2, and DES SV data sets. Our preliminary
results agree with the Planck constraints assuming the Lambda-CDM model.
Overall, this thesis forges an innovative tool for future WL surveys. The
manuscript provides a brief review on WL peak counts.

Cosmology and Nongalactic Astrophysics