  Weak gravitational lensing, the deflection of light by mass, is one of the
best tools to constrain the growth of cosmic structure with time and reveal the
nature of dark energy. I discuss the sources of systematic uncertainty in weak
lensing measurements and their theoretical interpretation, including our
current understanding and other options for future improvement. These include
long-standing concerns such as the estimation of coherent shears from galaxy
images or redshift distributions of galaxies selected based on photometric
redshifts, along with systematic uncertainties that have received less
attention to date because they are subdominant contributors to the error budget
in current surveys. I also discuss methods for automated systematics detection
using survey data of the 2020s. The goal of this review is to describe the
current state of the field and what must be done so that if weak lensing
measurements lead toward surprising conclusions about key questions such as the
nature of dark energy, those conclusions will be credible.
