With more recent discoveries of oil and gas reserves in the deep ocean locations like Guyana and Ghana, floating vessels such as floating production storage and offloading (FPSO) and floating liquified natural gas (FLNG) are in high demand. Good seakeeping characteristics of floating vessels are relevant because they are expected to be in operation all year round regardless of the weather conditions they may encounter. One phenomenon that affects the motion responses of floating vessels in severe weather conditions is sloshing in the cargo tanks. Vessel wave responses and sloshing may, individually and combined, cause damaging and dangerous motions. The availability of fast and accurate techniques for predicting and analyzing the motions and tank behavior in extreme weather conditions plays a significant role in the design and operations of floating vessels. Over the years, investigations have been carried out on the hydrodynamics of vessel motions, sloshing as a separate phenomenon as well as coupled vessel motions with sloshing analysis. This study reviews the existing techniques that are applicable for analyzing coupled vessel motions and sloshing in the tanks of floating vessels moored offshore. The pros and cons of each technique have been discussed, with the aim to help future researchers and engineers select the most appropriate method for design and analysis. This paper also identifies methods that are yet to be fully applied for coupled seakeeping—sloshing analysis.