NEW YORK (AP) — The presidential candidates have been laying out their visions of America's role in the world today -- while also taking subtle political jabs at each other.

Their foreign policy speeches in New York have been shaped by the violent protests taking place in the Middle East, prompted in part by an anti-Islamic video produced in the United States.

Obama told the United Nations that the violence in Libya that took the lives of the U.S. ambassador and three others were "attacks on America" -- and he called on world leaders to help confront the root causes of rage across the Muslim world.

Romney, speaking at Bill Clinton's gathering of global leaders, called the assault on the U.S. consulate in Benghazi a terrorist attack -- language that Obama himself hasn't used, but that his spokesman and secretary of state have used.

Romney said, "We somehow feel that we are at the mercy of events, rather than shaping events."

In his U.N. speech, Obama didn't refer to the video that surfaced last week of Romney telling donors that he didn't have much faith in peace prospects between Israelis and Palestinians. But Obama said, "The future must not belong to those who turn their backs on the prospect of peace."

Romney told the Clinton gathering that his foreign policy would encourage open markets and free enterprise in exchange for U.S. aid.

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149-a-11-(President Obama, addressing the United Nations General Assembly)-"and every faith"-President Obama says the anti-Muslim film that sparked deadly violence is 'crude and disgusting.' (25 Sep 2012)

<<CUT *149 (09/25/12)>> 00:11 "and every faith"

148-w-36-(Mark Smith, AP White House correspondent, with President Obama)--President Obama is telling the world's Muslims that no amount of anger over an offensive video justifies violence against U.S. interests. AP White House Correspondent Mark Smith reports. (25 Sep 2012)

<<CUT *148 (09/25/12)>> 00:36

150-a-15-(President Obama, addressing the United Nations General Assembly)-"hold in common"-President Obama says recent attacks in the Mideast are an assault on the United Nations' very ideals. (25 Sep 2012)

<<CUT *150 (09/25/12)>> 00:15 "hold in common"

161-a-13-(President Obama, addressing the United Nations General Assembly)-"to do so"-President Obama calls on leaders in the Muslim world to stand up for free speech, and he adds a personal note. (25 Sep 2012)

<<CUT *161 (09/25/12)>> 00:13 "to do so"

162-a-08-(President Obama, addressing the United Nations General Assembly)-"violence and extremism"-President Obama says world leaders must join to confront the root causes of rage in the Muslim world. (25 Sep 2012)

<<CUT *162 (09/25/12)>> 00:08 "violence and extremism"

134-a-12-(Mitt Romney, Republican presidential nominee, addressing the Clinton Global Initiative's annual meeting)-"America's own economy"-Republican presidential nominee Mitt Romney says more of America's foreign aid should be dedicated to promoting work and free enterprise. (25 Sep 2012)

<<CUT *134 (09/25/12)>> 00:12 "America's own economy"

135-a-11-(Mitt Romney, Republican presidential nominee, addressing the Clinton Global Initiative's annual meeting)-"money runs out"-Republican presidential nominee Mitt Romney says while a temporary aid package can jolt an economy, that's not enough. (25 Sep 2012)

<<CUT *135 (09/25/12)>> 00:11 "money runs out"

137-a-16-(Mitt Romney, Republican presidential nominee, addressing the Clinton Global Initiative's annual meeting)-"nuclear weapons capability"-Republican presidential nominee Mitt Romney says he's troubled by recent events in the Mideast. (25 Sep 2012)

<<CUT *137 (09/25/12)>> 00:16 "nuclear weapons capability"

133-w-30-(Mark Smith, AP White House correspondent, with Mitt Romney, Republican presidential nominee)--Mitt Romney is calling for a shift in U.S. development assistance from aid to trade. AP White House Correspondent Mark Smith reports. (25 Sep 2012)

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136-a-11-(Mitt Romney, Republican presidential nominee, addressing the Clinton Global Initiative's annual meeting)-"and reward entrepreneurs"-Republican presidential nominee Mitt Romney says he has a new approach to foreign aid. (25 Sep 2012)

<<CUT *136 (09/25/12)>> 00:11 "and reward entrepreneurs"

APPHOTO NYML109: Republican presidential candidate, former Massachusetts Gov. Mitt Romney is introduced by former President Bill Clinton at the Clinton Global Initiative, Tuesday, Sept. 25, 2012, in New York. (AP Photo/Mark Lennihan) (25 Sep 2012)

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APPHOTO NYML107: Republican presidential candidate, former Massachusetts Gov. Mitt Romney speaks at the Clinton Global Initiative, Tuesday, Sept. 25, 2012, in New York. (AP Photo/Mark Lennihan) (25 Sep 2012)

<<APPHOTO NYML107 (09/25/12)>>

APPHOTO UNJM110: President Barack Obama departs after addressing the 67th United Nations General Assembly, at U.N. headquarters, Tuesday, Sept. 25, 2012. (AP Photo/John Minchillo) (25 Sep 2012)

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APPHOTO UNSW119: President Barack Obama addresses the 67th session of the General Assembly at United Nations headquarters, Tuesday, Sept. 25, 2012. (AP Photo/Seth Wenig) (25 Sep 2012)

<<APPHOTO UNSW119 (09/25/12)>>