ABSTRACT We investigate the stochastic gravitational-wave background (SGWB) produced by merging binary black holes (BBHs) and binary neutron stars (BNSs) in the frequency ranges of Laser Interferometer Gravitational-Wave Observatory (LIGO)/Virgo/Kagra and Laser Interferometer Space Antenna (LISA). We develop three analytical models, which are calibrated to the measured local merger rates, and complement them with three population synthesis models based on the cosmic code. We discuss the uncertainties, focusing on the impact of the BBH mass distribution, the effect of the metallicity of the progenitor stars, and the time delay distribution between star formation and compact binary merger. We also explore the effect of uncertainties in binary stellar evolution on the background. For BBHs, our analytical models predict ΩGW in the range [4 × 10−10 to 1 × 10−9] (25 Hz) and [1 × 10−12 to 4 × 10−12] (3 mHz), and between [2 × 10−10 to 2 × 10−9] (25 Hz) and [7 × 10−13 to 7 × 10−12] (3 mHz) for our population synthesis models. This background is unlikely to be detected during the LIGO/Virgo/Kagra O4 run, but could be detectable with LISA. We predict about 10 BBH and no BNS mergers that could be individually detectable by LISA for a period of observation of 4 yr. Our study provides new insights into the population of compact binaries and the main sources of uncertainty in the astrophysical SGWB.