3,7-Disubstituted-2 or 3-cephem-4-carboxylic acid compounds of the formula: ##STR1## wherein R.sup.1 is acylamino or ar(lower)alkylamino; R.sup.2 is carboxy or carboxy protected by a pharmaceutically acceptable group; --N--R.sup.3, R.sup.4 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 or hydroxy(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 ##STR2## and pharmaceutically acceptable salts thereof. The present compounds are useful for the treatment of infectious diseases in animals and human beings.