When humans listen to speech, their neural activity tracks the slow amplitude fluctuations of the speech signal over time, known as the speech envelope. Studies suggest that the quality of this tracking is related to the quality of speech comprehension. However, a critical unanswered question is how envelope tracking arises and what role it plays in language development. Relatedly, its causal role in comprehension remains unclear, as some studies have found it to be present even for unintelligible speech. Using electroencephalography, we investigated whether the neural activity of newborns and 6-month-olds is able to track the speech envelope of familiar and unfamiliar languages in order to explore the developmental origins and functional role of envelope tracking. Our results show that amplitude and phase tracking take place at birth for familiar and unfamiliar languages alike, i.e. independently of prenatal experience. However, by 6 months language familiarity modulates the ability to track the amplitude of the speech envelope, while phase tracking continues to be universal. Our findings support the hypothesis that amplitude and phase tracking could represent two different neural mechanisms of oscillatory synchronisation and may thus play different roles in speech perception.