The two-state solution would establish an independent Palestinian state alongside Israel — two states for two peoples. In theory, this would win Israel security and allow it to retain a Jewish demographic majority (letting the country remain Jewish and democratic) while granting the Palestinians a state.
Most governments and world bodies have set achievement of the two-state solution as official policy, including the United States, the United Nations, the Palestinian Authority and Israel. This goal has been the basis of peace talks for decades.
Why is the solution so difficult to achieve?
There are four issues that have proved most challenging. Each comes down to a set of bedrock demands between the two sides that, in execution, often appear to be mutually exclusive.
1. Borders: There is no consensus about precisely where to draw the line. Generally, most believe the border would follow the lines before the Arab-Israeli war of 1967, but with Israel keeping some of the land where it has built settlements and in exchange providing other land to the Palestinians to compensate. Israel has constructed barriers along and within the West Bank that many analysts worry create a de facto border, and it has built settlements in the West Bank that will make it difficult to establish that land as part of an independent Palestine. As time goes on, settlements grow, theoretically making any future Palestinian state smaller and possibly breaking it up into noncontiguous pieces.
2. Jerusalem: Both sides claim Jerusalem as their capital and consider it a center of religious worship and cultural heritage. The two-state solution typically calls for dividing it into an Israeli West and a Palestinian East, but it is not easy to draw the line — Jewish, Muslim and Christian holy sites are on top of one another. Israel has declared Jerusalem its “undivided capital,” effectively annexing its eastern half, and has built up construction that entrenches Israeli control of the city.
3. Refugees: Large numbers of Palestinians fled or were expelled from their homes in what is now Israel, primarily during the 1948 Arab-Israeli war that came after Israel’s creation. They and their descendants now number five million and believe they deserve the right to return. This is a nonstarter for Israel: Too many returnees would end Jews’ demographic majority and therefore Israel’s status as both a Jewish and a democratic state.
4. Security: For Palestinians, security means an end to foreign military occupation. For Israelis, this means avoiding a takeover of the West Bank by a group like Hamas that would threaten Israelis (as happened in Gaza after Israel’s 2005 withdrawal). It also means keeping Israel defensible against foreign armies, which often means requiring a continued Israeli military presence in parts of the West Bank.
Why do some consider the two-state solution dead?
There is plenty of blame to go around. The Palestinian leadership is divided between two governments that cannot come to terms. The leadership in the West Bank lacks the political legitimacy to make far-reaching but necessary concessions, and the leadership in Gaza does not even recognize Israel, whose citizens it frequently attacks. The United States, which has brokered talks for years, has taken more than a few missteps.
And most important, the current Israeli leadership, though it nominally supports a two-state solution, appears to oppose it in practice.
Benjamin Netanyahu, the Israeli prime minister since 2009, endorsed the two-state solution in a speech that year. But he continued to expand West Bank settlements and, in 2015, said there would be “no withdrawals” and “no concessions.”
Mr. Netanyahu appears personally skeptical of Palestinian independence. His fragile governing coalition also relies on right-wing parties that are skeptical of or outright oppose the two-state solution.
Israeli public pressure for a peace deal has declined. The reasons are complex: demographic changes, an increasingly powerful settler movement, outrage at Palestinian attacks such as a recent spate of stabbings, and bitter memories of the Second Intifada in the early 2000s, which saw frequent bus and cafe bombings.
And the status quo has, for most Israelis, become relatively peaceful and bearable. Many see little incentive for adopting a risky and uncertain two-state solution, leaving Mr. Netanyahu with scant reason to risk his political career on one.
Are there other solutions?
There are, but they involve such drastic costs that the United States and many other governments consider all but the two-state solution unacceptable.
There are multiple versions of the so-called one-state solution, which would join all territories as one nation. One version would grant equal rights to all in a state that would be neither Jewish nor Palestinian in character, because neither group would have a clear majority. Skeptics fear this would risk internal instability or even a return to war.
Another, advocated by some on the Israeli far right, would establish one state but preserve Israel’s Jewish character by denying full rights to Palestinians. Under this version, Israel would no longer be a democratic state.
With few viable or popular alternatives, the most likely choice may be to simply maintain the status quo — though few believe that is possible in the long term.
What happens if there is no solution?
A common prediction, as Mr. Kerry stated, is that Israel will be forced to choose between the two core components of its national identity: Jewish and democratic.
This choice, rather than coming in one decisive moment, would probably play out in many small choices over a process of years. For instance, a 2015 poll by the Israel Democracy Institute found that 74 percent of Jewish Israelis agreed that “decisions crucial to the state on issues of peace and security should be made by a Jewish majority.” That pollster also found that, from 2010 to 2014, Jewish Israelis became much less likely to say that Israel should be “Jewish and democratic,” with growing factions saying that it should be democratic first or, slightly more popular, Jewish first.
Many analysts also worry that the West Bank government, whose scant remaining legitimacy rests on delivering a peace deal, will collapse. This would force Israel to either tolerate chaos in the West Bank and a possible Hamas takeover or enforce a more direct form of occupation that would be costlier to both parties.
This risk of increased suffering, along with perhaps permanent setbacks in the national ambitions of both Palestinians and Israelis, is why Nathan Thrall, a Jerusalem-based analyst with the International Crisis Group, told me last year, “Perpetuating the status quo is the most frightening of the possibilities.”