We study packings of graphs with given maximal degree. We shall prove that the (hitherto unproved) Bollobás–Eldridge–Catlin Conjecture holds in a considerably stronger form if one of the graphs is d-degenerate for d not too large: if and then a d-degenerate graph of maximal degree and a graph of order n and maximal degree pack. We use this result to show that, for d fixed and n large enough, one can pack arbitrary d-degenerate n-vertex graphs of maximal degree at most .