The term "spin caloritronics" defines a novel branch of spintronics that focuses on the interplay between electron spins with heat currents. In the frame of this research area, this thesis is aimed at investigating the effect of a heat current on magnetization dynamics in two different typologies of systems and materials: magnetic insulators and metallic nanostructures. In the first case we conduct studies on yttrium iron garnet (YIG) samples subjected to a temperature gradient. The irreversible thermodynamics of a continuous medium with magnetic dipoles predicts that a thermal gradient across a YIG slab, in the presence of magnetization waves, produces a magnetic field that is the magnetic analog of the well known Seebeck effect. This thermally induced field can influence the time evolution of the magnetization, in such a way that it is possible to modulate the relaxation of the precession when applying a heat current. We found evidence for such a magnetic Seebeck effect (MSE) by conducting transmission measurements in a thin slab of YIG subjected to an in-plane temperature gradient. We showed how the MSE can modulate the magnetic damping depending on the direction of the propagating magnetostatic modes with respect to the orientation of the temperature gradient. In the second part of the thesis we focus our investigation on metallic nanostructures subjected to a heat current. In a metal, the three-current model (current of entropy, of spin up and spin down electrons) predicts that a heat current induces a spin current which will then influence the magnetization dynamics like a charge-driven spin current would. Hence, we explore what has been called Thermal Spin Torque in electrodeposited Co / Cu / Co asymmetric spin valves placed in the middle of copper nanowires. These samples are fabricated by conventional electrodeposition technique in porous polycarbonate membranes using an original method that allows high frequency electrical measurements. We used a modulated laser to investigate the effect of a temperature gradient. We observed a heat-driven spin torque by measuring electrically the quasi-static magnetic response of a spin valve when subjected to the heat current, generated by two laser diodes heating the electrical contact at one end or the other of the nanowire. Analysing the variation in the resistance induced by a heat-driven spin torque, represented by peaks occurring in correspondence with the GMR transition, we found that a temperature difference of the order of 5 K is sufficient to produce sizeable torque in spin valves.