We consider motion of overdamped Brownian particles in anisotropic, periodic structures (ratchets), that are driven by nonequilibrium fluctuations with finite correlation time (colored noise). Fluctuation-induced transport in such correlation ratchets which are driven by both, additive Gaussian white and additive Ornstein-Uhlenbeck noise is studied. As a second example we substitute the Ornstein-Uhlenbeck process by Harmonic Noise, which yields a richer structure for directed transport. For constant intensity noise, we depict the current versus the noise parameters. For Harmonic noise, the current exhibits (multiple) current reversals, while for Ornstein-Uhlenbeck noise such a current reversal emerges only if the shape of the ratchet potential is chosen appropriately. Moreover, we investigate the detailed behavior of the current carrying multidimensional stationary probability. Apart from maxima reflecting locally stable states, the probability assumes additional colored noise induced maxima. The colored noise induced, directed current is evaluated with the help of matrix continued fractions. This latter method is elucidated here in greater detail, covering also three-dimensional Fokker-Planck structures. These ratchet devices can be utilized to efficiently separate particles of nearly equal mass in nano- and microstructures.