The spin Seebeck effect refers to the generation of spin voltage as a result of a temperature gradient in ferromagnetic or ferrimagnetic materials. When a conductor is attached to a magnet under a temperature gradient, the thermally generated spin voltage in the magnet injects a spin current into the conductor, which in turn produces electric voltage owing to the spin-orbit interaction. The spin Seebeck effect is of increasing importance in spintronics, since it enables direct generation of a spin current from heat and appears in a variety of magnets ranging from metals and semiconductors to insulators. Recent studies on the spin Seebeck effect have been conducted mainly in paramagnetic metal/ferrimagnetic insulator junction systems in the longitudinal configuration in which a spin current flowing parallel to the temperature gradient is measured. This 'longitudinal spin Seebeck effect' (LSSE) has been observed in various sample systems and exclusively established by separating the spin-current contribution from extrinsic artefacts, such as conventional thermoelectric and magnetic proximity effects. The LSSE in insulators also provides a novel and versatile pathway to thermoelectric generation in combination of the inverse spin-Hall effects. In this paper, we review basic experiments on the LSSE and discuss its potential thermoelectric applications with several demonstrations.