The Gelb era has been transformative, but also tumultuous. A record executive before joining the Met in 2006, he began the company’s successful program of cinema simulcasts, which now shows operas on 2,200 screens in more than 70 countries across six continents. He appointed Mr. Nézet-Séguin, the Met’s first new music director in over four decades. And he has worked to modernize the company by bringing in new stage directors and 81 new productions. There have been 23 Met premieres during his tenure, of both neglected older works and newer ones by composers including John Adams, Philip Glass, Kaija Saariaho and Nico Muhly.

But amid industrywide struggles, Mr. Gelb has also faced fiscal pressures and weak box office revenues, which prompted him to curb expenses, causing occasional labor strife. He had to navigate the case of James Levine, the company’s former music director, who was fired in 2018 after an inquiry the Met commissioned in response to news reports found evidence that Mr. Levine had engaged in sexual misconduct, which he denied. Mr. Levine’s subsequent lawsuit against the Met, and the Met’s countersuit, were quietly settled in August.

When Plácido Domingo, long one of the Met’s biggest stars, was accused in news reports this summer of sexually harassing multiple women, Mr. Gelb initially said he would await the results of an inquiry into the matter before taking action. But after some members of the Met’s company expressed concerns about that approach, the Met asked Mr. Domingo to withdraw from its fall production of “Macbeth” — which he did, adding that he did not expect to ever return.

The financial challenges facing the Met were underscored last week when the news that it had run modest deficits for the past two years — including a $1.1 million deficit on a budget of $312 million in the 2019 fiscal year — prompted S & P Global Ratings to announce that it was keeping the company’s “A” credit rating but revising its outlook to negative, from stable.

“There is still a lot of work to be done in terms of finding the formula for economic stability and sustainability,” Mr. Gelb said.