This invention relates to new cephem and cepham compounds of high antimicrobial activity of the formula: ##STR1## wherein R.sup.1 is amino or a protected amino,R.sup.2 is hydrogen, acyl, aryl which may be substituted with suitable substituent(s), lower alkyl substituted with suitable substituent(s), lower alkenyl, lower alkynyl, cycloalkyl which may be substituted with suitable substituent(s), cyclo(lower)alkenyl, or S or O containing 5-membered heterocyclic group substituted with oxo group(s),R.sup.3 is hydrogen or lower alkyl,R.sup.4 is hydrogen, acyloxy(lower)alkyl; acylthio(lower) alkyl; pyridinium(lower)alkyl which may be substituted with suitable substituent(s); a heterocyclicthio(lower)alkyl which may be substituted with suitable substituent(s); lower alkyl;l halogen; hydroxy; thiazolium(lower)alkyl which may be substituted with suitable substituent(s); or lower alkoxy; R.sup.5 is carboxy or a protected carboxy, wherein R.sup.5 is COO.sup.- when R.sup.4 is pyridinium(lower)alkyl which may be substituted with suitable substituent(s) or thiazolium(lower)alkyl which may be substituted with suitable substituent(s), and the heavy solid line means single or double bond; and pharmaceutically acceptable salt thereof.