One of the actual challenges of spintronics is the realization of a spin transistor allowing control of spin transport through an electrostatic gate. In this paper, we report on different experiments which demonstrate gate control of spin transport in a carbon nanotube connected to ferromagnetic leads. We also discuss some theoretical approaches which can be used to analyse spin transport in these systems. We emphasize the roles of the gate-tunable quasi-bound states inside the nanotube and the coherent spin-dependent scattering at the interfaces between the nanotube and its ferromagnetic contacts.