The genesis of the work on this family of distributions was the paper by Burr 1942, in which he aimed to generate distributions that could take on a wide variety of shapes and yet remain tractable to work with. Subsequent work with the family has been sporadic and has concentrated on the univariate case. It is also spread over a wide range of disciplines. Here we survey the literature concerning the Burr family of distributions and summarise and extend the results concerning two members of the family, the Burr types II and XII. We also develop some new univariate and multivariate Burr type II distributions, which being a generalisation of the Logistic, may prove useful in future applications.