A first-generation (i.e. translate-rotate) micro X-ray transmission computed tomography system has been developed, which utilizes a standard 2.2 kW long-fine-focus diffraction tube with Cu-anode as the X-ray source, a spherical W/C multilayer-mirror to condense and spectrally select the CuKα-radiation (8.04 keV) from the tube and a scintillation counter to detect the X-ray photons; in the present configuration the optical system demagnifies the original source size in the direction parallel to the imaged object slice by a factor of 5, where a small slit captures the radiation and thus gives an intense microscopic (pseudo-) source of monochromatic X-radiation in close vicinity of the scanned specimen. The system provides tomographic images of small objects (up to 25 mm in diameter) reconstructed as 128 × 128 matrices with resolutions between ∼ 20 and 200 μm in ≥ 10 min. The software package which is available for image reconstruction includes filtered backprojection, correcting backprojection (ART, MART) and a new type of weighted backprojection, which turns out to be a simplified version of MART (SMART). A dedicated scan- and reconstruction-procedure demonstrates the feasibility to image selected regions-of-interest within the investigated specimen slice with (up to 1 order of magnitude) higher spatial resolution than their surroundings without major artefacts (Zoom-CT). The hard-and software-components of this CT-system are discussed, several examples are given and perspectives of further development are outlined.