By Jonathan Marcus
BBC Diplomatic Correspondent
Mr Netanyahu has first call to try to build an Israeli government
Likud leader Benjamin Netanyahu, who was Israeli Prime Minister during the 1990s, now has the chance of a significant political comeback.
But the immediate question is not so much can he form a coalition government, but rather what sort of government will it be?
Mr Netanyahu likes to see himself as a man of destiny.
He is an admirer of the great British wartime leader Winston Churchill.
And, to paraphrase Churchill, this is not the end of the coalition-building process in Israel but rather only the end of the beginning.
Stage one is over. Mr Netanyahu has first call to try to build a government.
Ranged to the right of his own Likud formation he has a variety of parties including the Russian immigrant-backed Yisrael Beiteinu; orthodox Jewish religious parties like Shas; and the ultra-nationalist splinter formations from the break-up of the National Religious Party.
These together could give him a majority in the 120-seat Knesset, or parliament.
He would first, of course, have to reconcile Shas with the stridently secular Yisrael Beiteinu.
The leader of Yisrael Beitenu, Avigdor Lieberman, is a key coalition figure
Yisrael Beiteinu calls for the introduction of civil marriage in Israel and for the end to what it sees as the religious parties' blackmail of the Israeli political system.
But what would a government of the right and ultra-right be formed to do?
It clearly would not be there to make concessions to the Palestinians. There would be strong pressures from within such a coalition to go full steam ahead on settlement-building on the West Bank.
Such a government could easily find itself on a collision course with the Obama administration in Washington.
Most Israeli commentators believe that Mr Netanyahu's goal is still to build as broadly-based a coalition as possible.
This is clearly what the Israeli President Shimon Peres would prefer. And it is what Mr Netanyahu himself has said he wants to do.
He has called upon both the Kadima leader Tzipi Livni and the Labour Party head, Ehud Barak, to join him in a broad, national unity coalition.
Analysts discount claims from Kadima or Labour leaders that they are heading into opposition as so much rhetoric.
What really matters now is what jobs are on offer and what policies a national unity government might pursue.
Unity on Iran
The challenges ahead are enormous.
For a start there is the economic crisis. Unemployment is rising, Israel is being battered by the same economic forces that are hitting the wider global economy and urgent action is needed.
On the security front there is still the unfinished business of the Gaza ceasefire.
One theme has emerged clearly: the need... to be ready to tackle the Iran problem at source
There is also the stuttering Annapolis peace process between Israel and Fatah and the need to craft a good relationship with the new team in Washington.
But looming over everything else is the question of Iran and its nuclear programme.
Mr Netanyahu mentioned this specifically when accepting the president's call to form a new government.
Israel's fragmented and feuding political class can be accused of many things: an absence of vision; of narrow ambition; and a failure of statesmanship.
But one issue and one issue alone draws out the statesman in all of them, and that is Iran.
In numerous discussions with Israeli analysts and commentators over recent days one theme has emerged clearly: the need, perhaps within a year or 18 months, to be ready to tackle the Iran problem at source.
This issue unites politicians across the political spectrum. The Obama administration will be given ample time to pursue its diplomatic opening towards Tehran.
But if it fails Israel is going to be ready to act militarily.
To do this the country needs a stable government. And it needs a government that is not so ideologically-driven that it inevitably clashes with Washington at every turn.
Mr Netanyahu now has a job on his hands. But the real horse-trading is only just beginning.