Mesoscale eddies, which are widely found in the oceans, play a vital role in momentum, energy, heat and mass transport. The Euler method for identifying mesoscale eddies using satellite altimeter data was presented in detail, including closed SLA contours, OW numbers, Winding-Angle and flow vector methods. The results show that mesoscale eddies are almost nonlinear and solid-body rotation. The long-lived eddies with lifetimes ≥16 weeks have an average lifetime of 32 weeks and an average propagation distance of 550 km. Their mean amplitude and a speed-based radius scale as defined by the automated procedure are 8 cm and 90 km, respectively. The method combining with SLA and Argo profiles to composite the three-dimensional structure were addressed. Due to the different temperature and salt structure in the various oceans of the world, the eddies in different oceans show different three-dimensional structures, which are influenced by both the generation and local temperature and salinity. Two special types of eddies were introduced from the perspective of generation, evolution and dissipation processes, namely the Loop Current Ring in the Gulf of Mexico and Mediterranean eddy in the Atlantic Ocean. Finally, issues including submesoscale processes, dissipation of eddies and subthermocline eddies were discussed, and some future research directions were proposed.