7-Amino-2 or 3-cephem-4-carboxylic acid compounds of the formula: ##STR1## wherein R.sup.1 is amino; R.sup.2 is carboxy or carboxy protected by a pharmaceutically acceptable group; ##STR2## is di(lower)alkylamino(lower)alkylamino, an unsaturated or saturated 3 to 8 membered heteromonocyclic(lower)alkylamino group containing 1 to 4 nitrogen atoms, an unsaturated or saturated 3 to 8 membered heteromonocyclic(lower)alkylamino group containing 1 to 4 nitrogen atoms substituted by lower alkyl orhydroxy(lower)alkyl, an unsaturated or saturated 5 to 6 membered heteromonocyclic group containing 2 to 4 nitrogen atoms, an unsaturated or saturated 5 to 6 membered heteromonocyclic group containing 2 to 4 nitrogen atoms substituted by lower alkyl or hydroxy(lower)alkyl or hydroxypiperidino, and X is --S or ##STR3## and pharmaceutically acceptable salts thereof.