(ABRIDGED) We describe the first results of the ALHAMBRA survey which provides cosmic tomography of the evolution of the contents of the Universe over most of Cosmic history. Our approach employs 20 contiguous, equal-width, medium-band filters covering from 3500 to 9700 A, plus the JHKs bands, to observe an area of 4 sqdeg on the sky. The optical photometric system has been designed to maximize the number of objects with accurate classification by SED and redshift, and to be sensitive to relatively faint emission lines. The observations are being carried out with the Calar Alto 3.5m telescope using the cameras LAICA and O-2000. The first data confirm that we are reaching the expected magnitude limits of AB<~25 mag in the optical filters from the blue to 8300 A, and from AB=24.7 to 23.4 for the redder ones. The limit in the NIR is (Vega) K_s~20, H~21, J~22. We expect to obtain accurate redshift values, Delta z/(1+z) <~ 0.03 for about 5x10^5 galaxies with I<~25 (60% complete), and z_med=0.74. This accuracy, together with the homogeneity of the selection function, will allow for the study of the redshift evolution of the large scale structure, the galaxy population and its evolution with redshift, the identification of clusters of galaxies, and many other studies, without the need for any further follow-up. It will also provide targets for detailed studies with 10m-class telescopes. Given its area, spectral coverage and its depth, apart from those main goals, the ALHAMBRA-Survey will also produce valuable data for galactic studies.
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