Being gravitational in origin, weak lensing (WL) effects are uniquely important in mapping the cosmic large-scale structures that encode important information of the dark side of the universe, namely dark matter and dark energy. The enormous advancement of large astronomical surveys over the past two decades has made WL a major probe that has delivered cosmological information with comparable precision and great complementarity as those derived from other probes. This article aims to present a brief overview about the current status of WL cosmology and the targeted high precision achievable with the upcoming Stage IV surveys. Challenges to meet the stringent accuracy requirements that match the unprecedented precision will also be discussed.