The PETRA IV project aims at upgrading the present synchrotron radiation source PETRA III at DESY into an ultralow-emittance source. Being diffraction limited up to X-rays of about 10 keV, PETRA IV will be ideal for three-dimensional X-ray microscopy of biological, chemical and physical processes under realistic conditions at length scales from atomic dimensions to millimetres and time scales down to the sub-nanosecond regime. In this way, it will enable groundbreaking studies in many fields of science and industry, such as health, energy, earth and environment, mobility and information technology. The science case is reviewed and the current state of the conceptual design is summarized, discussing a reference lattice, a hybrid multi-bend achromat with an interleaved sextupole configuration based on the ESRF-EBS design, in more detail as well as alternative lattice concepts.