In this review we present a progress report of the connection between long-duration gamma-ray bursts (GRBs) and their accompanying supernovae (SNe). The analysis is from the point of view of an observer, with much of the emphasis placed on how observations, and the modelling of observations, have constrained what we known about GRB-SNe. We discuss their photometric and spectroscopic properties, their role as cosmological probes, including their measured luminosity$-$decline relationships, and how they can be used to measure the Hubble constant. We present a statistical analysis of their bolometric properties, and use this to determine the properties of the "average" GRB-SNe: which has a kinetic energy of $E_{\rm K} ≈ 2.5\times10^{52}$ erg, an ejecta mass of $M_{\rm ej} ≈ 6$ M$_{⊙}$, a nickel mass of $M_{\rm Ni} ≈ 0.4$ M$_{⊙}$, a peak photospheric velocity of $v_{\rm ph} ≈ 21,000$ km s$^{-1}$, a peak bolometric luminosity of $L_{\rm p} ≈ 1\times10^{43}$ erg s$^{-1}$, and it reaches peak bolometric light in $t_{\rm p} ≈ 13$ days. We discuss their geometry, consider the various physical processes that are thought to power the luminosity of GRB-SNe, and whether differences exist between GRB-SNe and the SNe associated with ultra-long duration GRBs. We discuss how observations of the environments of GRB-SNe further constrain the physical properties of their progenitor stars, and give an overview of the current theoretical paradigms of their suspected central engines. We also present an overview of the radioactively powered transients that have been photometrically associated with short-duration GRBs. We conclude the review by discussing what additional research is needed to further our understanding of GRB-SNe, in particular the role of binary-formation channels and the connection of GRB-SNe with superluminous SNe (abridged). ; Comment: Accepted review in the special issue on "GRB in the Swift/Fermi Era and Beyond" published by Advances in Astronomy