The fitting of radial velocity curves is a frequent procedure in binary star and exoplanet research. In the majority of cases, the fitting routines need to be fed with a set of initial parameter values and priors from which to begin the computations and their results can be affected by local minima. We present a new code, the rvfit code, for fitting radial velocities of stellar binaries and exoplanets using an adaptive simulated annealing (ASA) global minimization method, which quickly converges to a global solution minimum without the need to provide preliminary parameter values. We show the performance of the code using both synthetic and real datasets: double-lined binaries, single-lined binaries, and exoplanet systems. In all examples, the Keplerian orbital parameters fitted by the rvfit code and their computed uncertainties are compared with literature solutions. Finally, we provide the source code, with a working example and a detailed description of how to use it.