The SCUBA 8-mJy survey is the largest submillimetre (submm) extragalactic mapping survey undertaken to date, covering 260 arcmin2 to a 4 σ detection limit of ≃8 mJy at 850 μm, centred on the Lockman Hole and ELAIS N2 regions. Here, we present the results of new 1.4-GHz imaging of these fields, of the depth and resolution necessary to reliably identify radio counterparts for 18 of 30 submm sources, with possible detections of a further 25 per cent. Armed with this greatly improved positional information, we present and analyse new optical, near-infrared (near-IR) and XMM–Newton X-ray imaging to identify optical/IR host galaxies to half of the submm-selected sources in those fields. As many as 15 per cent of the submm sources detected at 1.4 GHz are resolved by the 1.4-arcsec beam and a further 25 per cent have more than one radio counterpart, suggesting that radio and submm emission arise from extended starbursts and that interactions are common. We note that less than a quarter of the submm-selected sample would have been recovered by targeting optically faint radio sources, underlining the selective nature of such surveys. At least 60 per cent of the radio-confirmed optical/IR host galaxies appear to be morphologically distorted; many are composite systems – red galaxies with relatively blue companions; just over one half are found to be very red (I − K > 3.3) or extremely red (I − K > 4); contrary to popular belief, most are sufficiently bright to be tackled with spectrographs on 8-m telescopes. We find one submm source which is associated with the steep-spectrum lobe of a radio galaxy, at least two more with flatter radio spectra typical of radio-loud active galactic nuclei (AGN), one of them variable. The latter is amongst four sources (≡15 per cent of the full sample) with X-ray emission consistent with obscured AGN, though the AGN would need to be Compton thick to power the observed far-IR luminosity. We exploit our well-matched radio and submm data to estimate the median redshift of the S850μm ∼ 8 mJy submm galaxy population. If the radio/far-IR correlation holds at high redshift, and our sample is unbiased, we derive a conservative limit of 〈z〉 ≥2.0, or ≥2.4 using spectral templates more representative of known submm galaxies.