We present new deep co-adds of data taken within Stripe 82 of the Sloan Digital Sky Survey (SDSS), especially stacked to reach the faintest surface brightness limits of this data set. Stripe 82 covers 275 square degrees within -50{deg}<=RA<=+60{deg} and -1.26{deg}<=DE<=+1.25{deg}. We discuss the steps of our reduction which puts special emphasis on preserving the characteristics of the background (sky + diffuse light) in the input images using a non-aggressive sky subtraction strategy. Our reduction reaches a limit of ~28.5 mag/arcsec^2^ (3{sigma}, 10x10 arcsec^2^) in the r band. The effective surface brightness limit (50 per cent completeness for exponential light distribution) lies at <{mu}e(r)>~25.5mag/arcsec^2^. For point sources, we reach 50 per cent completeness limits (3{sigma} level) of (24.2, 25.2, 24.7, 24.3, 23.0) mag in (u, g, r, i, z). This is between 1.7 and 2.0mag deeper than the single-epoch SDSS releases. The co-adds show point spread functions (PSFs) with median full width at half-maximum values ranging from 1-arcsec in i and z to 1.3-arcsec in the u band. The imaging data are made publicly available at http://www.iac.es/proyecto/stripe82. The release includes deep co-adds and representations of the PSF for each field. Additionally, we provide object catalogues with stars and galaxies confidently separated until g~23mag. The IAC Stripe 82 co-adds offer a rather unique possibility to study the low surface brightness Universe, exemplified by the discovery of stellar streams around NGC 0426 and NGC 0936. We also discuss further science cases like stellar haloes and disc truncations, low surface brightness galaxies, the intracluster light in galaxy clusters and the diffuse emission of Galactic dust known as Galactic Cirrus.