The new Lagrangian approach is developed to study transport, mixing, eddies and fronts in the ocean. It is based on calculation of relevant Lagrangian indicators which are functions of trajectories of virtual particles advected in the altimetry-based velocity field, outputs of numerical circulation models and reanalysis. In this approach, the relevant indicators of water motion are computed to provide information on the origin, advection history, 'age' and other properties of water masses. We review the achievements and limitations of the Lagrangian diagnostics. The Lagrangian approach allows us not only to identify, track real mesoscale eddies and to document 'biography' of each eddy anywhere in the ocean, both retrospectively and in the near-real time. In particular, we can estimate the fractions of water masses/types the eddy's cores consist of. Using this approach and automatic eddy detection algorithm, a census of the mesoscale eddies in the northwestern Pacific have been done in the altimetry era. The ocean fronts are characterized by a confluence of waters with different properties. These features promote aggregation of nutrients, phytoplankton and zooplankton contributing to creation of oases of marine life in the oceanic desert for marine habitants, from small pelagic fish to seabirds and top predators. We have applied this approach for capturing frontal features based on extraction of Lagrangian fronts in chaotic oceanic flows with the focus on their role for fisheries. The location of Lagrangian fronts can be calculated in the real time with satellite-derived velocity fields under any weather conditions and in the areas with small contrasts of sea surface temperature. The proximity of fishing ground for different species to location of the fronts is shown with the help of statistical tests in different seas and oceans based on numerous catch reports of fishing vessels. These results may help in avoiding the risk of overfishing and bycatch. We also emphasize the importance of fronts in marine ecology, sustainable fisheries and in making decisions on the organization of marine protected areas. Transport barriers in the oceanic flows are considered from the point of view of dynamical systems theory. Special attention is paid to discussion of importance of transport barriers in some practical issues such as anthropogenic (oil spills, radionuclides etc.) and natural pollution.