We present an RR Lyrae (RRL) catalogue based on the combination of the third data release of the Zwicky Transient Facility (ZTF DR3) and Gaia EDR3. We use a multistep classification pipeline relying on the Fourier decomposition fitting to the multiband ZTF light curves and random forest classification. The resulting catalogue contains 71755 RRLs with period and light-curve parameter measurements and has a completeness of 0.92 and a purity of 0.92 with respect to the Specific Objects Study Gaia DR2 RRLs. The catalogue covers the Northern sky with declination => -28{deg}, its completeness is ~> 0.8 for heliocentric distance <= 80 kpc, and the most distant RRL is at 132 kpc. Compared with several other RRL catalogues covering the Northern sky, our catalogue has more RRLs around the Galactic halo and is more complete at low-Galactic latitude areas. Analysing the spatial distribution of RRL in the catalogue reveals the previously known major overdensities of the Galactic halo, such as the Virgo overdensity and the Hercules-Aquila Cloud, with some evidence of an association between the two. We also analyse the Oosterhoff fraction differences throughout the halo, comparing it with the density distribution, finding increasing Oosterhoff I fraction at the elliptical radii between 16 and 32 kpc and some evidence of different Oosterhoff fractions across various halo substructures.