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The Murchison Widefield Array (MWA) has observed the entire southern sky (Declination, 
$\delta< 30^{\circ}$
) at low radio frequencies, over the range 72–231MHz. These observations constitute the GaLactic and Extragalactic All-sky MWA (GLEAM) Survey, and we use the extragalactic catalogue (EGC) (Galactic latitude, 
$|b| >10^{\circ}$
) to define the GLEAM 4-Jy (G4Jy) Sample. This is a complete sample of the ‘brightest’ radio sources (
$S_{\textrm{151\,MHz}}>4\,\text{Jy}$
), the majority of which are active galactic nuclei with powerful radio jets. Crucially, low-frequency observations allow the selection of such sources in an orientation-independent way (i.e. minimising the bias caused by Doppler boosting, inherent in high-frequency surveys). We then use higher-resolution radio images, and information at other wavelengths, to morphologically classify the brightest components in GLEAM. We also conduct cross-checks against the literature and perform internal matching, in order to improve sample completeness (which is estimated to be 
$>95.5$
%). This results in a catalogue of 1863 sources, making the G4Jy Sample over 10 times larger than that of the revised Third Cambridge Catalogue of Radio Sources (3CRR; 
$S_{\textrm{178\,MHz}}>10.9\,\text{Jy}$
). Of these G4Jy sources, 78 are resolved by the MWA (Phase-I) synthesised beam (
$\sim2$
arcmin at 200MHz), and we label 67% of the sample as ‘single’, 26% as ‘double’, 4% as ‘triple’, and 3% as having ‘complex’ morphology at 
$\sim1\,\text{GHz}$
(45 arcsec resolution). We characterise the spectral behaviour of these objects in the radio and find that the median spectral index is 
$\alpha=-0.740 \pm 0.012$
between 151 and 843MHz, and 
$\alpha=-0.786 \pm 0.006$
between 151MHz and 1400MHz (assuming a power-law description, 
$S_{\nu} \propto \nu^{\alpha}$
), compared to 
$\alpha=-0.829 \pm 0.006$
within the GLEAM band. Alongside this, our value-added catalogue provides mid-infrared source associations (subject to 6” resolution at 3.4
$\mu$
m) for the radio emission, as identified through visual inspection and thorough checks against the literature. As such, the G4Jy Sample can be used as a reliable training set for cross-identification via machine-learning algorithms. We also estimate the angular size of the sources, based on their associated components at 
$\sim1\,\text{GHz}$
, and perform a flux density comparison for 67 G4Jy sources that overlap with 3CRR. Analysis of multi-wavelength data, and spectral curvature between 72MHz and 20GHz, will be presented in subsequent papers, and details for accessing all G4Jy overlays are provided at https://github.com/svw26/G4Jy.
The entire southern sky (Declination, 
$\delta< 30^{\circ}$
) has been observed using the Murchison Widefield Array (MWA), which provides radio imaging of 
$\sim$
2 arcmin resolution at low frequencies (72–231 MHz). This is the GaLactic and Extragalactic All-sky MWA (GLEAM) Survey, and we have previously used a combination of visual inspection, cross-checks against the literature, and internal matching to identify the ‘brightest’ radio-sources (
$S_{\mathrm{151\,MHz}}>4$
Jy) in the extragalactic catalogue (Galactic latitude, 
$|b| >10^{\circ}$
). We refer to these 1 863 sources as the GLEAM 4-Jy (G4Jy) Sample, and use radio images (of 
${\leq}45$
arcsec resolution), and multi-wavelength information, to assess their morphology and identify the galaxy that is hosting the radio emission (where appropriate). Details of how to access all of the overlays used for this work are available at https://github.com/svw26/G4Jy. Alongside this we conduct further checks against the literature, which we document here for individual sources. Whilst the vast majority of the G4Jy Sample are active galactic nuclei with powerful radio-jets, we highlight that it also contains a nebula, two nearby, star-forming galaxies, a cluster relic, and a cluster halo. There are also three extended sources for which we are unable to infer the mechanism that gives rise to the low-frequency emission. In the G4Jy catalogue we provide mid-infrared identifications for 86% of the sources, and flag the remainder as: having an uncertain identification (129 sources), having a faint/uncharacterised mid-infrared host (126 sources), or it being inappropriate to specify a host (2 sources). For the subset of 129 sources, there is ambiguity concerning candidate host-galaxies, and this includes four sources (B0424–728, B0703–451, 3C 198, and 3C 403.1) where we question the existing identification.
