The LUXE experiment aims at studying high-field QED in electron-laser and photon-laser interactions, with the 16.5 GeV electron beam of the European XFEL and a laser beam with power of up to 350 TW. The experiment will measure the spectra of electrons, positrons and photons in expected rates from $10^{-3}$ to $10^{9}$ particles per 1 Hz bunch crossing, depending on the laser power and focus. These measurements have to be performed in the presence of low-energy high radiation background. To meet these challenges, for high-rate electron and photon fluxes, the experiment will use Cherenkov radiation detectors, scintillator screens, sapphire sensors as well as lead-glass monitors for backscattering off the beam dump. A four-layer silicon-pixel tracker and a compact electromagnetic tungsten calorimeter with silicon sensors will be used to measure the positron spectra. The layout of the experiment and the expected performance under the harsh radiation conditions are discussed. Beam tests results for the Cherenkov detector and the electromagnetic calorimeter prototypes are presented.