We prove that Gaussian continuous-variable quantum key distribution protocols, using a Gaussian distribution of coherent or squeezed states and homodyne or heterodyne measurement, are secure against arbitrary attacks. Our proof exploits the specific symmetries in phase-space of Gaussian QKD protocols to prove that once a simple test over the measurement outcomes succeeds, the global state shared between Alice and Bob is well decribed by assigning a low dimensional Hilbert space to each mode. Then one can use the postselection technique introduced by Christandl, Koenig and Renner for discrete-variable protocols to conclude. Our result greatly improves over previous ones using either a de Finetti theorem or an entropic uncertainty principle which could not be applied to prove the security of protocols in realistic experimental implementations.