The weekend's election result has the Liberal Party on the hunt for a new leader, with Peter Dutton emerging as the most likely replacement for Scott Morrison.
Albanese was sworn in with his deputy, Richard Marles, and frontbenchers Penny Wong, Jim Chalmers and Katy Gallagher.
Marles was sworn in as Employment Minister, Senator Wong as Foreign Affairs Minister, Chalmers as Treasurer and Gallagher as Finance Minister, Minister for Women and Attorney-General.
He and Senator Wong will fly to Japan later today to meet with the leaders of the Quad - the United States, India and Japan.
"Travelling to the Quad meeting in week one signifies how important we believe this partnership is for our security," Senator Wong said in a statement.
"And we will be taking new energy and much more to the table - including our commitment to act on climate change after a lost decade."
US President Joe Biden phoned Albanese late on Sunday, congratulating him on the win and thanking him for choosing to travel to Tokyo for the meeting.
The five politicians will hold all the portfolios until Albanese swears in his full frontbench.
Chalmers is the new Treasurer and Senator Gallagher the Finance Minister.
There has been uncertainty about what role Marles, the incoming deputy prime minister, will hold. As deputy leader, he can pick his portfolio and there is speculation he wants to return to defence.
Albanese wants national cabinet to meet in person in the coming weeks as he sets about implementing his agenda.
Dutton emerging as next Liberal leader
Liberal sources have told the ABC that Dutton, who served as a senior cabinet minister throughout the Coalition's three terms, has emerged as the frontrunner to replace Morrison.
Treasurer Josh Frydenberg was among the biggest victims of the Coalition's loss, losing both his seat and the likely leadership of the Liberals in opposition.
The ABC has been told by sources that, without Frydenberg, Dutton appeared to have the numbers to become leader.
Sources have also told the ABC that South Australian senator Anne Ruston is being touted as a possible deputy to Dutton, with Liberals conceding they need to do more to get women into their ranks.
About half the Liberal women in the lower house lost their seats in this election, fuelling the need for a woman to play a senior role in the leadership.
Cabinet ministers Karen Andrews and Sussan Ley's names have also been touted, but their support in the party room remains unclear.
Bridget Archer, who retained her seat - despite it being the Coalition's most marginal - told the ABC on Monday morning that she was considering nominating to be deputy leader, vowing to be a vocal, moderate voice.
The Liberal Party's greatest losses came thanks to teal independents - women who wiped out moderate Liberals from once-safe seats in Sydney, Melbourne and Perth.
Liberal sources expect Frydenberg will mount a political comeback but his path back to the parliament remains unclear.
"I still have fire in my belly," he said on Sunday morning, while refusing to concede until postal votes were counted.
New parliament brings mixed Labor fortunes
Labor has lost two women from its cabinet ranks, with Kristina Keneally's attempt to be parachuted into a safe Western Sydney seat failing, and Terri Butler losing her Brisbane seat to the Greens.
The Greens will enter the next parliament with record numbers in their ranks. The party will hit a high-water mark of 12 in the Senate, where it now holds the balance of power.
Thanks to gains in Queensland, the party has also added to its lower house seats.
Labor had a big night in Perth, wiping out Coalition cabinet ministers and once-safe, prized Liberal seats.
Attorney-General Michaelia Cash, the deputy Liberal leader in the Senate, blamed Morrison for the party's fortunes in the west.
She said his decision to initially back Clive Palmer's legal action against WA's closed border proved costly, with Morrison's close confidant Ben Morton losing his Perth seat.
"We - Senator Cash and Mr Morton - did not support siding with Clive Palmer but the decision was made that we would," she said.