WASHINGTON—President Donald Trump on Sunday issued a new ban on entry to the U.S. that applies a range of restrictions on nationals from eight countries, including new targets Chad, North Korea and Venezuela.

The new restrictions, replacing his previous order that expired on Sunday, apply to five of the six countries covered by the previous travel ban—Iran, Libya, Somalia, Syria and Yemen. The African nation of Sudan was subject to the original ban but dropped under the new version.

The new rules take effect on Oct. 18. Countries that were subject to an earlier version will operate under the old rules until then.

The new rules vary by country, with some nations facing outright travel bans and others subject to more limited restrictions. For instance, nationals of some countries are barred from migrating to the U.S. but are allowed to travel for school. Still, in most cases, the majority of potential travel to the U.S. will be banned.

“As president, I must act to protect the security and interests of the United States and its people,” Mr. Trump wrote in the Sunday order.