An antibacterial cephalosporin hydroxamic acid derivative represented by the following formula (I), processes for its preparation, compositions containing the same as an active ingredient, and a method for killing bacteria by contacting with said compound. ##STR1## (wherein, R is acyl; R.sup.1 is hydrogen or methoxy; R.sup.2 is alkylene; R.sup.3 is hydrogen, aliphatic group or a hydroxy-protecting group; R.sup.4 is hydrogen or amino-protecting group; R.sup.5 is hydrogen, light metal atom or carboxy-protecting group; and X is oxygen, sulfur, or sulfinyl).