Albright, who had cancer, died in March at the age of 84.
She served as secretary of state from 1997 to 2001 under President Bill Clinton after serving as the U.S. ambassador to the U.N. from 1993 to 1997.
President Joe Biden eulogized Albright, sharing a story of a speech he gave last month in Poland, where he said a crowd of hundreds cheered when Albright's name was mentioned.
"Her name is still synonymous with America as a force for good in the world. Madeleine never minced words or wasted time when she saw something needed fixing, or someone who needed helping. She just got to work," Biden said.
During her tenure as secretary of state, she focused on promoting the eastward expansion of NATO and pushed for NATO intervention in the 1999 war in Kosovo, according to the historical office of the Department of State.
MORE: Madeleine Albright, first female US secretary of state, dies at 84
Her approach to diplomacy and statecraft was colored by her own experiences as a refugee who fled what was then Czechoslovakia with her family in the aftermath of World War II.
She remained engaged with both American and international affairs until the end of her life, writing a book in 2018 warning about a resurgence of fascism and sounding an alarm about Russian President Vladimir Putin in a New York Times op-ed published just before Russia's invasion of Ukraine.
"In early 2000, I became the first senior U.S. official to meet with Vladimir Putin in his new capacity as acting president of Russia... Flying home, I recorded my impressions. 'Putin is small and pale,' I wrote, 'so cold as to be almost reptilian,'" Albright wrote in the Times. She added that "should he invade [Ukraine], it will be a historic error."
Both former President Bill Clinton and former Secretary of State Hillary Clinton were also slated to speak at the funeral, according to the Washington National Cathedral.
"Few leaders have been so perfectly suited for the times in which they served... Because she knew firsthand that America's policy decisions had the power to make a difference in people's lives around the world, she saw her jobs as both an obligation and an opportunity," the former president wrote in a statement the day Albright died.
Her daughters, Anne, Alice and Katie, were also scheduled to speak.
ABC News' Emily Shapiro contributed to this report.