The emporers that followed Augustus (Tiberius, Caligula, Claudius, Nero) are referred to as the Julio-Claudians.
Augustus' greatest challenge seems to have been securing the succession.
The reasons Augustus struggled to secure the succession were; he had no natural son and therefore needed to secure one through adoption, he had recurring ill health, his long life meant that he kept outliving candidates, and to secure the prestige of the family line.
The most difficult aspect was the construction of the principate meant that a position of ruler could not be inherited.
There were several ways he attempted to solve the problem of succession:
by associating family members with tribunitian potestas for a certain term
also associating family members with power of imperium
by fast promotion in the cursus honorum by reducing age limits
by using marriage to try and promote heirs within the family (through Julia) and making alliances with those who could act in their interest
Marcellus:
nephew
Augustus made him marry Julia
served with Tiberius in Cantabrian wars
allowed to be consul for 10 years
23BC - fell ill, didn't recover and died the same year
Agrippa:
best friend and son in law
architect of Rome's building programme
granted Tribunician power in 18 and 13 BC
proconsulare imperium in Augustus provinces
died 12BC, Augustus mourned for a month and held a lavish funeral
Tiberius:
step son
distinguished himself as praetor and on campaigns in Germania
consul in 13BC
forced to marry Julia
exiled himself to Rhodes, Augustus refused his return
allowed to return in 4AD
Gaius and LuciusCaesar:
grandsons
princeps iuventutis, leaders of youth
both made consul designate at 15
Gaius active in politics, undertook expedition to Arabia
Armenian leader, trapped and murdered Gaius in Asia Minor
Lucius died after a short illness
Agrippa Posthumus:
grandson
participated in a series of games for 'equestrian youth'
not adopted alongside Gaius and Lucius
Augustus considered him to have a 'beastly nature'