Some of the most exciting scientific challenges at present in cosmology are in understanding the nature of dark matter and dark energy. The billions of galaxies observed by the Legacy Survey of Space and Time (LSST) at Rubin Observatory will dramatically improve the statistical power of weak lensing observations and help probe the mass distribution in the universe more accurately. This increased statistical sensitivity means that potential systematic biases must be carefully identified, quantified, and minimized. This thesis addresses two such systematic biases: galaxy color gradients and blending.The first part of the thesis describes how shape measurements of galaxies with a varying spectral energy distribution across their profile — called "color gradients" -- when impacted by a chromatic point spread function can be biased. The expected bias is estimated for the LSST using simulations of parametric galaxies and realistic galaxy images. The predicted multiplicative shear biases due to color gradients are found to be at least a factor of two below the LSST full-depth requirement on the total systematic uncertainty in the redshift-dependent shear calibration. The second part of the thesis focuses on the blending challenge for the LSST where a significant fraction of the lensed galaxy images will overlap with images of other objects, affecting the accuracy of flux and shape measurements. Two novel approaches to infer the presence of objects that go undetected because of blending are described and their performances are compared to existing detection algorithms.