Traits in any organism are not independent, but show considerable integration, observed in a form of couplings and trade-offs. Therefore, improvement in one trait may affect other traits, often in undesired direction. To account for this problem, crop breeding increasingly relies on multi-trait genomic prediction (MT-GP) approaches that leverage the availability of genetic markers from different populations along with advances in high-throughput precision phenotyping. While significant progress has been made to jointly model multiple traits using a variety of statistical and machine learning approaches, there is no systematic comparison of advantages and shortcomings of the existing classes of MT-GP models. Here, we fill this knowledge gap by first classifying the existing MT-GP models and briefly summarizing their general principles, modeling assumptions, and potential limitations. We then perform an extensive comparative analysis with ten traits measured in an Oryza sativa diversity panel using cross-validation scenarios relevant in breeding practice. Finally, we discuss directions that can enable the building of next generation MT-GP models in addressing pressing challenges in crop breeding.