Monoclonal antibodies are attractive therapeutics for treating a wide range of human disorders due to their exquisite binding specificity and high binding affinity. However, a limitation of antibodies is their highly variable and difficult-to-predict propensities to aggregate when concentrated during purification and delivery. Despite the large size and complex structure of antibodies, recent findings suggest that antibody solubility can be dramatically improved using rational design methods in addition to conventional selection methods. Here, we review key advances and unmet challenges in engineering the variable and constant regions of antibody fragments and full-length antibodies to resist aggregation without reducing their binding affinity. These experimental and computational discoveries should accelerate the development of robust algorithms for designing aggregation-resistant antibodies.