MATISSE is the mid-infrared spectrograph and imager for the Very Large Telescope Interferometer (VLTI) at Paranal. This second generation interferometry instrument will open new avenues in the exploration of our Universe. Mid-infrared interferometry with MATISSE will allow significant advances in various fundamental research fields: studies of disks around young stellar objects where planets form and evolve, surface structures and mass loss of stars in late evolutionary stages, and the environments of black holes in active galactic nuclei. MATISSE is a unique instrument. As a first breakthrough it will enlarge the spectral domain used by optical interferometry by offering the L & M bands in addition to the N band, opening a wide wavelength domain, ranging from 2.8 to 13 μm on angular scales of 3 mas (L/M band) / 10 mas (N band). As a second breakthrough, it will allow mid-infrared imaging – closure-phase aperture-synthesis imaging – with up to four Unit Telescopes (UT) or Auxiliary Telescopes (AT) of the VLTI. MATISSE will offer various ranges of spectral resolution between R~30 to ~5000. In this article, we present some of the main science objectives that have driven the instrument design. We introduce the physical concept of MATISSE including a description of the signal on the detectors and an evaluation of the expected performance and discuss the project status. The operations concept will be detailed in a more specific future article, illustrating the observing templates operating the instrument, the data reduction and analysis, and the image reconstruction software.