Ehum, techically it did in periods, but what you are asking assumes the existence of sons. The first two imperial dynasties were families, the Julio-Claudians and the Flavians. Have a look:
Augustus had no sons of his own but adopted several (who died) until he was left with adopting his wife's son Tiberius as his heir. Tiberius didn't have any sons either, so the nephew of Tiberius - Gaius (Caligula) - was chosen as emperor. He didn't last for very long in the imperial seat and was assassinated after 4 years, only to be replaced by his uncle, Claudius. He didn't have any sons of his own either, and so adopted his great-nephew Nero as heir.

The story of Nero is well known (although the truth can be discussed) so it is easy to understand why Rome didn't want to continue on with the poisonous Julio-Claudian family. After Nero the year of 4 emperors followed, after which Vespasianus of the Flavian dynasty became lasting emperor. He actually had two sons - Titus and Domitianus. Titus was his chosen heir and even co-ruled with him for a short while before Vespasianus died, leaving Titus as emperor. However, Titus only lived for another 2 years after that, meaning that the title instead fell to Domitianus. Domitianus was one of the worst emperors Rome ever had (a bit subjective, but a general consensus agrees that he was bad) and although he ruled for over 2 decades before his assassination, he did nothing that made Rome consider continuing on with the Flavian dynasty. A dynasty that started off as immensely popular with such strong emperors as divus Vespasianus and divus Titus, who constructed the Flavian amphitheatre (Colosseum) together.

These two dynasties were followed by "the adoptive emperors", called so for obvious reasons. Nerva, who was made emperor after the assassination of Domitianus (much because of his age), had no sons and chose Trajanus as his adoptive heir. He too was left without sons of his own and thus had to adopt Hadrianus as his heir. See the pattern? Hadrianus didn't have any sons of his own either, and some circumstancial stories suggest that he was gay, and adopted two sons. The former died, but the latter - Antoninus Pius - became emperor. Like with the previous ones, Antoninus didn't get any sons (well, adult such that is) and thus adopted Marcus Aurelius as his heir, as well as asking him to marry his daughter. Essentially making him marry his sister. Anyway, Marcus Aurelius actually had a son of his own - Commodus - who, like Domitianus, is considered to have been an awful emperor.

With the assassination of Commodus came a new era of Roman politics where the imperial guard had indirect absolute power. They had had such ever since the ascension of Caligula, but after Commodus' death they started to control who became emperor. And with that the custom of father to (adoptive) son heritage ended in large. Septimus Severus, who came to rule as the first of "the Soldier emperors", actually fathered two sons and decided that they would rule Rome together. However, they both hated each other and Caracalla ordered Geta killed - probably a good decision since it is likely that Geta was planning the same for his brother. Caracalla was the like of Domitianus and Commodus and really not fit to have any power at all, let alone rule an empire. After his death, years followed of imperial power passing to the guy with the most military influence in the empire.

Which is a good place to end this text. Apologies for the rambling sort of text, it kind of follows that an answer to a question about heir becomes like that. Anyway, I hope this answers your question.