LIVE: Syriza projected to win Greek elections
The TV exit polsl would give Tsipras 30-34% of the vote, and centre-right New Democracy 28.5-32.5%, but who comes first is particularly important in Greek politics, since the winner gets an extra 50 seat bonus in parliament.
The radical left-wing party was catapulted into power in January this year, in opposition to the country's European bailout deals and austerity policies.
Eight months later, after months of torturous negotiations, capitulation, the resignation of former finance minister Yanis Varoufakis and the calling of new snap elections, Prime Minister Alexis Tsipras is now heading up a pro-bailout party.
The anti-bailout hard left of Syriza are running under a separate banner, as a bloc called Popular Unity.
The polls have generally given Syriza a narrow lead, but it's been uncomfortably close to New Democracy, which headed up the country's coalition government until early this year.
Here's what the polls are saying for other parties:
No party is anywhere near an absolute majority, so whoever wins will have the difficult task of forming a coalition, probably with centrist and centre-left parties like To Potami and Pasok. Tsipras has expressed a preference for a coalition with the Independent Greeks (ANEL) but it's not clear that they'll get back into parliament.
Here's how the seats would look based on the exit polls:
Whichever party wins, a lot of what comes ahead will be a question of interpretation. Both Tsipras and ND leader Evangelos Meimarakis are committed to bring in the country's third bailout deal.
We'll have more as the results come out.