The novel five-Penning-trap mass spectrometer PENTATRAP is developed at the Max-Planck-Institut für Kernphysik (MPIK), Heidelberg. Ions of interest are longlived highly charged nuclides up to bare uranium. PENTATRAP aims for an accuracy of a few parts in 1012 for mass ratios of mass doublets. A physics program for PENTATRAP includesQ-value measurements of ß-transitions relevant for neutrino physics, stringent tests of quantum electrodynam- ics in the regime of extreme electric fields, and a test of special relativity. Main features of PENTATRAP are an access to a source of highly charged ions, a multi-trap configuration, simultaneous measurements of frequencies, a continuous precise monitoring of magnetic field fluctuations, a fast exchange between different ions, and a highly sensitive cryogenic non-destructive detection system. This article gives a motivation for the new mass spectrometer PENTATRAP, presents its experimental setup, and describes the present status.