• President Trump and Kim Jong-un, the leader of North Korea, failed to reach a deal on denuclearization during their second summit meeting in eight months. “Sometimes you have to walk,” Mr. Trump said after the talks broke down.

• Mr. Trump said Mr. Kim was willing to close some but not all nuclear sites in North Korea in exchange for the lifting of all international sanctions.

• The leaders were scheduled to have lunch together and sign a joint agreement, but those plans were abruptly called off.