The gens Claudia was one of the oldest families in ancient Rome, and for centuries its members were regularly leaders of the city and empire.
The family was traditionally held to have begun with Attius Clausus, a Sabine who favored peace with Rome, an unpopular position that led to him leaving Regillus with his followers around 504 BC. Rome was welcoming however, making his followers citizens and giving them land, and making Appius Claudius Sabinus (as his name was said in the Roman dialect) a senator.
Clodius was a "plebeian" spelling used by some members of the gens, while Claudia (and Clodia) were the forms used by women.
Appius Claudius Sabinus, founder
Appius Claudius Crassus(?), decemvir ca 450 BC, consul 445 BC
Caius Claudius, consul 454(?)
Appius Claudius P.f. Crassus Inregillensis, consul 346 BC
M. Claudius C.f. Marcellus, consul 329 BC
Appius Claudius Caecus, consul 307 BC, 297 BC (or 296 BC)
M. Claudius M.f. Marcellus, consul 288 BC
C. Claudius M.f. Canina, consul 286 BC, 274 BC
C. (or K.) Quinctius L.f. Claudus, consul 272 BC
Ap. Claudius Ap.f. Russus, consul 269 BC
Appius Claudius Caudex, consul 265 BC
Publius Claudius Pulcher, consul 250 BC
C. Claudius Ap.f. (Caecus f.?) Centho, consul 240 BC
Marcus Claudius Marcellus, consul 222 BC, suffect 215 BC, 214 BC, 210 BC, 208 BC
Quintus Claudius, tribune 218 BC
Ap. Claudius P.f. Pulcher, consul 212 BC
C. Claudius Ti.f. Nero, consul 207 BC
Ti. Claudius P.f. Nero, consul 202 BC
M. Claudius M.f. Marcellus, consul 196 BC
Ap. Claudius Ap.f. Pulcher, consul 185 BC
P. Claudius Ap.f. Pulcher, consul 184 BC
M. Claudius M.f. Marcellus, consul 183 BC
Gaius Claudius Pulcher, consul 177 BC
M. Claudius Marcellus, consul 166 BC, 155 BC, 152 BC
Appius Claudius Pulcher, consul 144 BC (or 143 BC)
Gaius Claudius Pulcher, consul 92 BC
Appius Claudius Pulcher, consul 77 BC (or 79 BC?)
Quintus Claudius Quadrigarius, historian
Appius Claudius Pulcher, consul 54 BC
Marcus Claudius Marcellus, consul 51 BC
Gaius Claudius Marcellus, consul 51 BC, 50 BC
Appius Claudius Pulcher, consul 38 BC
Marcus Claudius Marcellus Aeserninus, consul 22 BC
Nero Claudius Drusus, consul 9 BC
Tiberius Claudius Nero, father of the emperor Tiberius
Tiberius Claudius Nero Caesar (Tiberius), emperor
Tiberius Claudius Nero Germanicus (Claudius I), emperor
Claudius Nero Caesar (Nero), emperor
Claudius Ptolemaeus (Ptolemy), Greek astronomer
Claudius Civilis, led a revolt in 69
L. Catilius Severus Iulianus Claudius Reginus, consul 120
M. Gavius Claudius Squilla Gallicanus, consul 127
L. Uibullius Hipparchus Ti. Claudius Atticus Herodes, consul 143
Cn. Claudius Severus Arabianus, consul 146
Cn. Claudius Severus, consul 173
Ti. Claudius Pompeianus, consul 173
Maternus Ti. Claudius, consul 185
Ti. Claudius Seuerus Proculus, consul 200
Ap. Claudius Iulianus, consul 224
Claudius Pompeianust, consul 231
Cn. Claudius Seuerus, consul 235
L. Ti. Claudius Aurelius Quintianus, consul 235
Claudius Aelianus (Aelian)
Claudius Galenus (Galen), Greek physician
Claudius II ("Gothicus"), emperor
Marcus Claudius Tacitus, emperor
T. Claudius M. Aurelius Aristobulus, consul 285
Flauius Claudius Constantinus Caesar (Constantine II), emperor
Flavius Claudius Julianus (Julian), emperor
Claudius Mamertinus, consul 362
Sex. Claudius Petronius Probus, consul 371
Flauius Claudius Antonius, consul 382
Claudius Claudianus (Claudian), poet
Imp. Caesar Flauius Claudius Constantinus Augustus (Constantine III), emperor
Claudius Iulius Eclesius Dynamius, consul 488
Claudius Salmasius is "Claude Saumaise" Latinized, as was formerly common for European scholars.