We report on the design and performance of a velocity map imaging (VMI) spectrometer optimized for experiments using high-intensity extreme ultraviolet (XUV) sources such as laser-driven high-order harmonic generation (HHG) sources and free-electron lasers (FELs). Typically exhibiting low repetition rates and high single-shot count rates, such experiments do not easily lend themselves to coincident detection of photo-electrons and -ions. In order to obtain molecular frame or reaction channel-specific information, one has to rely on other correlation techniques, such as covariant detection schemes. Our device allows for combining different photo-electron and -ion detection modes for covariance analysis. We present the expected performance in the different detection modes and present the first results using an intense high-order harmonic generation (HHG) source. of their impact coordinates on the detector, as well as from the flight time, acquired by means of, e.g., delay-line detectors. REMI gives access to complete, correlated, 3D velocity information of all particles detected in coincidence, as long as the count rates are sufficiently low (<1 event/shot) to avoid detecting fragments from more than one target atom or molecule on the same shot.With high-intensity sources, the repetition rates are typically low, while the single-shot count rates can be very high, which in many cases makes it difficult to use techniques relying on detecting coincidences. Another approach is to use the so-called velocity map imaging (VMI) technique [21], which uses an extraction field configuration that makes the impact coordinates on the detector independent of the location of the ionization event within the interaction volume, as well as of the momentum along the detector axis. This allows for the use of a micro-channel plate (MCP) and a phosphor screen where the impact of a large number of particles can be accumulated on every shot. Under the condition of cylindrical symmetry of the ionization process, the initial three-dimensional momentum distribution of the particles can be recovered from the measured two-dimensional projection using numerical inversion procedures [22,23].There are several demonstrations of using VMI, under low count rate conditions, for photoelectron-photoion coincidence spectroscopy (PEPICO) [24], for example using electron VMI combined with ion time-of-flight (TOF), ion VMI together with electron spectroscopy and electron VMI with ion VMI [25][26][27][28][29].An obvious drawback of VMI in high count rate conditions is that one cannot rely on coincidence detection for extracting information about different electrons or ions coming from the same target molecule. An elegant way to overcome this lack of correlated information in VMI, without sacrificing the high count rates, is to use covariance mapping [30]. Briefly, for any two variables, X ], sampled synchronously in a repetitive measurement, one can calculate the covariance, which is a measure of how well correlated the variations of the two variables are. C...