During World War II, President Franklin D. Roosevelt selected Sidney J. Weinberg, the chief executive of Goldman Sachs, to serve as the assistant director of the War Production Board. Mr. Weinberg was one of the few men on Wall Street to support Roosevelt in the 1932 election, and he went on to advise Presidents Harry S. Truman, Dwight D. Eisenhower and Lyndon B. Johnson.

Known as “Mr. Wall Street,” Mr. Weinberg began a tradition of Goldman executives taking leading roles in public service that is continuing under President Trump.

Goldman Sachs alumni also have influence in other branches of government and overseas. Mario Draghi, the president of the European Central Bank, was vice chairman for Goldman Sachs in Europe. And many current and previous senators and representatives spent years at the influential Wall Street firm.

Here is a look at some of the firm’s ties during recent administrations:

The Trump Administration

Stephen K. Bannon, chief strategist to the president and a former Goldman banker

Mr. Bannon is most closely associated with Breitbart News, the website that helped become an outlet for his right-wing, populist outlook that is suspicious of the established corporate and government elite.

But he was a Goldman Sachs mergers and acquisitions banker in the 1980s before he left to start his own boutique investment firm. There, he advised on a deal for the production company behind “Seinfeld,” and is said to have been paid for his services via cash as well as a share of the show’s profits.

Gary D. Cohn, director of the National Economic Council and Goldman Sachs president

Economic policy making will be influenced by the experiences of some key Goldman alumni in the Trump administration.

Mr. Cohn, a longtime second-in-command at Goldman Sachs, is now overseeing economic policy in the White House as director of the National Economic Council.

Steven T. Mnuchin, Treasury secretary and Goldman partner

Mr. Mnuchin, a financier with deep roots on Wall Street and in Hollywood but with no previous government experience, was the national finance chairman for Mr. Trump’s campaign. After leaving Goldman, he created his own hedge fund, moved to the West Coast and bankrolled hits like the “X-Men” franchise and “Avatar.”

James Donovan, nominee for deputy Treasury secretary and Goldman partner

Mr. Mnuchin’s pick to be his deputy also comes from Goldman Sachs. If confirmed, Mr. Donovan will play a substantial role in shaping some of the Trump administration’s biggest fiscal priorities, such as overhauling the tax code. Mr. Donovan has spent nearly 25 years at Goldman, where he worked with companies and individuals both as an investment banking adviser and as an investment management overseer.

Dina Powell, deputy national security adviser for strategy and longtime Goldman executive

Ms. Powell, who had been a senior counselor to the president, was named a deputy national security adviser for strategy this week. Although she does not have extensive experience on national security issues, Ms. Powell has spent 15 years in government.

Ms. Powell was recently the president of the Goldman Sachs Foundation. During her years at Goldman Sachs, she worked closely with Mr. Cohn.

The Obama Administration

William C. Dudley, partner at Goldman and president of the Federal Reserve Bank of New York

A Goldman economist with a knack for reading the markets, Mr. Dudley stepped into the spotlight in 2009 as the Fed’s senior statesman on Wall Street. He has navigated a painful recession while leading the Fed’s sweeping efforts to stabilize the nation’s troubled financial industry.

Gary Gensler, a partner at Goldman and chairman of the Commodity Futures Trading Commission

After minting a small fortune as one of the youngest partners in Goldman’s history, Mr. Gensler was named assistant secretary for financial markets in 1997 in the Clinton administration. At the Treasury Department, he helped enact legislation exempting broad portions of derivatives trading from oversight.

Under the Obama administration, Mr. Gensler developed a reputation for cracking down on Wall Street during his time at the agency. The day in 2010 that the Dodd-Frank financial overhaul became final, he stayed past 4 a.m. to put the finishing touches on the law. His aggressive streak thrust the once-backwater agency into the front lines of reform. The push by Mr. Gensler clashed with the staid culture of an agency once known as the “watchdog that didn’t bark.”

The George W. Bush Administration

Stephen Friedman, co-chairman of Goldman and White House chief economic adviser

Around Goldman, Mr. Friedman is remembered for having built the company’s lucrative practice of advising companies on mergers and acquisitions.

In 2002, President George W. Bush chose the tenacious Wall Street veteran as his chief economic adviser. Mr. Friedman’s co-chairman at Goldman, Robert E. Rubin, led President Clinton’s National Economic Council.

Mr. Friedman generated some controversy as chairman of the Federal Reserve Bank of New York during the financial crisis, when the Fed was helping put together a rescue plan for Wall Street. He stepped down from that role in 2009 after questions arose about his ties to Goldman.

Joshua B. Bolten, executive for Goldman Sachs International and White House chief of staff

The former Goldman executive grew up in establishment Washington, the son of a Central Intelligence Agency officer. Mr. Bolten was the White House legislative affairs director for part of the administration of the first President Bush.

Under the second President Bush, he was deputy White House chief of staff, then director of the Office of Management and Budget and finally chief of staff. Mr. Bolten played an important role in putting together the Bush tax plan and helped recruit another Goldman executive, Henry M. Paulson Jr., as Treasury secretary. In 2008, the House voted to issue contempt citations against Mr. Bolten and a former White House counsel for refusing to cooperate in an investigation into the mass firings of federal prosecutors.

Robert K. Steel, vice chairman of Goldman and under secretary of the Treasury

Before the financial crisis, Mr. Steel was co-chairman of one commission that claimed that heavy-handed regulation was hindering financial innovation and another that argued that hedge funds could police themselves. By April 2008, he was extolling the powers that a “superregulator” might wield over Wall Street one day.

“When you are driving fast down a slippery road, sometimes a regulator needs to tap lightly on the brakes to get you to slow down,” Mr. Steel told The New York Times. But to many on Wall Street and on Main Street, the car had already crashed.

Henry M. Paulson Jr., chief executive of Goldman and Treasury secretary

Before he became Treasury secretary in 2006, Mr. Paulson agreed to hold himself to a higher ethical standard than his predecessors. That plan did not survive the worst financial crisis since the Great Depression. The government propped up the teetering financial system with tens of billions of taxpayer dollars, including aid that directly benefited his former company. To deal with the financial crisis, Mr. Paulson tapped so many former Goldman executives that bankers coined the nickname “Government Sachs.”

“I operated very consistently within the ethic guidelines I had as secretary of the Treasury,” Mr. Paulson told lawmakers in 2009, adding that he asked for an ethics waiver for his interactions with his old company “when it became clear that we had some very significant issues with Goldman Sachs.”

Mr. Paulson helped decide the fates of a variety of financial companies, including two longtime Goldman rivals, Bear Stearns and Lehman Brothers, before his ethics waivers were granted. Ad hoc actions were taken by Mr. Paulson and officials at the Federal Reserve, like letting Lehman fail and compensating the trading partners of the American International Group.

Neel T. Kashkari, investment banker for Goldman and president of the Federal Reserve Bank of Minneapolis.

Mr. Kashkari arrived in Washington in 2006 after spending two years as a low-level technology investment banker for Goldman in San Francisco. Today, he runs the Federal Reserve Bank of Minneapolis and compares banking to the nuclear power industry.

The Clinton Administration

Robert E. Rubin, co-chairman of Goldman and Treasury secretary

Mr. Rubin amassed a fortune on Wall Street before heading the National Economic Council under President Bill Clinton. In 1994, Mr. Clinton selected him to become Treasury secretary. Mr. Rubin drew criticism just after his arrival in Washington when it was disclosed that he had sent farewell letters to hundreds of Goldman clients saying he was “looking forward to working with you in my new capacity.” He said that he was merely being polite.

Mr. Rubin played a role in arranging the compromise that led to the repeal of the Glass-Steagall Act. After leaving office, he joined a leadership triangle at Citigroup.

Kenneth D. Brody, management committee at Goldman and president of the Export-Import Bank

After running Goldman’s real estate division, Mr. Brody was appointed by President Clinton to the once-lethargic Ex-Im Bank. Critics have contended that the bank’s business is really a form of corporate welfare benefiting mainly giants like Boeing and General Electric.

When he arrived in Washington, he was astounded when an aide, told that it was difficult for the public to reach the agency on the telephone, replied that “those who need us know how to get us.” Mr. Brody said, “I went nuts” in response.