The 36-year-old says he is still available for selection while Cricket Australia will announce his replacement tomorrow.
Ricky Ponting steps down as Australia ODI and Test captain
The 36-year-old, who has led Australia in one-day internationals (ODI) since 2002 and Tests since 2004, had been under increasing pressure after an Ashes defeat to England and a quarter-final exit as the defending champions at the World Cup.
"I've had the chance to think long and hard about it and today I've decided to stand down as captain of the Test team and the one-day team as of now," he told a news conference at the Sydney Cricket Ground (SCG).
"The main reason for me is I think it's the right time.
"Getting through the Ashes series the way that we did and the World Cup now being over, I wanted to make sure that I gave the next captain every opportunity I possibly could to make sure he has as much experience going forward in the next couple of big events that we will play."
"All that being said I've made it very clear to selectors that I'm still available for selection in both forms of the games, so hopefully they will pick me and I'll be on a plane to Bangladesh in the next couple of days," added Ponting, referring to Australia's three ODIs against Bangladesh next month.
"I am keen to stay around and play a bit longer with the team where it is at the moment. I think I've got a lot to offer as a player and certainly as a leader for some of the younger players around, if and when required."
Cricket Australia declined to name a successor, but Michael Clarke, who stood in for the injured Ponting in the final Ashes Test and for the one-day series against England that followed it, is expected to replace his mentor as skipper.
Ponting endorsed the 29-year-old Clarke to take over the reins, adding that he would have no problems serving under his apprentice.
"I've actually taken orders most of my life, so it will be no different than anyone else here," Ponting said, smiling.
"I just want to do the best that I can for Australian cricket, for the next captain and for the team over the next period of time."
Cricket Australia are expected to name captains for Australia's Test, one-day and Twenty20 sides tomorrow.
Clarke stepped down from the Twenty20 captaincy following Australia's defeat in the Ashes in January, saying he wanted to concentrate on getting his batting form back. Cameron White assumed the role for two T20 matches against England in January.
Ponting reiterated his call that the selectors should resist calls to clear the decks of the national teams in a bid to blood new talent, with a number of senior players in their mid-30s.
"What I think you don't and you can't ever afford to have in the team is to have a mass exodus of all those sort of players at once," he said.
"I think we've seen that in the last couple of years when we had a lot of the greats move on at one time. It left us very bare. It's been hard for us to rebuild and get ourselves back to where we want to as a result of that."
While Ponting's batting record brooks no argument, his captaincy has often been criticised as Australia has declined as a force in world cricket, with three Ashes series defeats to England in particular damaging his reputation.
Australia went to the World Cup as double defending champions but returned home after defeat to India in the last eight, their unbeaten run of matches in the tournament having been ended at 34 by Pakistan in the group stage.
On his return to Australia at the weekend, Ponting acknowledged the pressure on him and said he would consider his position before the squad to tour Bangladesh was announced on Wednesday.
Ponting bucked a poor run of form to hit a defiant century in the quarter-final defeat to India in Ahmedabad, which will only have increased his determination to continue playing international cricket.
Despite former players and pundits calling for him to step down for months, Ponting said he had never been under any pressure from Australian selectors or administrators to relinquish the captaincy.
"I think if the decision was too late, I think I would have ... my bosses as in Cricket Australia telling me a while ago that they felt my time might have been up.
"I can go on record again and say that I've never received that sort of message."