The European Union’s new recommendation to halt nonessential travel from the U.S. due to the rise of Covid-19 cases stateside could create fresh virus-related travel uncertainty.

Monday’s announcement suggested that vaccinated travelers will still be permitted into EU member countries, though it is up to each nation to set their own restrictions. Stavros Lambrinidis, the EU’s ambassador to the U.S., said on Twitter that the recommendation means only essential travelers and vaccinated people from the U.S. would be allowed into the bloc.

The EU travel list is reviewed every two weeks and isn’t binding for member states, though it has generally set the pattern for who can visit the 27-country bloc. The EU had previously decided in June to add the U.S. to its “safe list.”

The European recommendation could thwart some Americans’ fall and winter trip plans, adding more challenges to an already complex year for international travel—one marked by passport delays, changing vaccination and testing requirements and myriad rules for booking at various destinations. 

Here’s what we know so far about what the EU recommendation means for travelers. 

Is the EU open to U.S. citizens?

Citing the spread of Covid-19 cases stateside, EU countries voted to remove the U.S. from a “safe list” of countries for nonessential travel, meaning vacations and recreational trips. The EU reviews its travel list every two weeks. Pressure to remove the U.S. from the travel list has also risen because Washington has maintained a ban on Europeans’ nonessential travel to the U.S.

Under EU rules, the bloc is supposed to consider removing from the safe list any country with more than 75 new Covid-19 cases per 100,000 inhabitants in the previous 14 days. The authority can take other factors into account when making its decision, including whether the third country has opened up to EU citizens.

The U.S. infection rate rose above 75 per 100,000 earlier in the summer, but EU member states agreed not to respond immediately. 

Member states have become frustrated in recent weeks over the U.S.’s refusal to drop restrictions on EU travelers. Earlier this month, European Commission President Ursula von der Leyen warned that the EU wouldn’t allow the lack of reciprocity to “drag on for weeks.”

Does this mean I can’t travel to Europe?

For now, travel is proceeding as normal. It isn’t yet clear which, if any, European countries would follow the recommendation. Each of the EU’s 27 member countries can decide whether they will close to nonessential U.S. travelers.

They may also choose to put restrictions only on unvaccinated travelers. When announcing the change, the EU said the decision doesn’t take away “the possibility for member states to lift the temporary restriction on nonessential travel…for fully vaccinated travelers.”

Will my airline refund my flight if I can’t travel?

Many carriers have eliminated change fees on domestic and international flights. As of Monday afternoon, airlines said they would continue to monitor the situation but provided few specifics.

“We continue to evaluate our travel waivers and policies based on market conditions and current travel restrictions that are in place,” said Andrea Koos, a spokeswoman for American Airlines. 

United Airlines also said it would keep customers informed of any changes to their travel plans. “We’ll continue to monitor how member states respond to this new guidance,” said spokeswoman Nicole Carriere.

Delta Air Lines said customers can use an interactive map on its website to review changing requirements. Morgan Durrant, a spokesman for the airline, also noted that if a flight is canceled for any reason—whether domestic or international—customers are entitled to a refund.

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Will my travel insurance cover my trip if the country I’m visiting isn’t open to visitors?

Most standard trip-cancellation policies won’t cover this type of situation, says Stan Sandberg, co-founder of TravelInsurance.com, a policy-comparison site. “The border shutdown is one of the big holes still in the coverage terms of most standard travel-insurance policies,” he says. 

Mr. Sandberg said that, in the beginning of the pandemic, some travelers were able to receive refunds from airlines and hotels when borders closed. “Of course, that still is dependent on the travel supplier, the hotel or the airline, and what their policies are,” he says.

Travelers who have “cancel for any reason” coverage will find themselves in a better position to get money back. This coverage is often sold as a supplemental policy and will allow you to receive reimbursement for your prepaid and nonrefundable trip costs. It can cost up to about 60% of your base insurance plan and will typically reimburse about 75% of your trip cost.

“So you’re not going to get 100% back,” Mr. Sandberg says. “That ‘cancel for any reason’ feature does give you the best protection against a border closing or government shutdown.”

What else should I keep in mind?

U.S. travelers have faced many complications traveling abroad this summer. The time frame to receive a passport is far longer than it was pre-pandemic, with standard passport applications taking up to 18 weeks and expedited applications taking up to 12 weeks. 

Additionally, many attractions across the EU, including museums, are requiring reservations this year to head off large gatherings.

Jenna Cannon, 33 years old, and her husband, are set to leave for their honeymoon in Greece on Friday. Should new rules interfere, it would be the fourth time they have had to cancel honeymoon plans since marrying in 2019.

Both Ms. Cannon and her husband, who live in the Baltimore area, are fully vaccinated and had planned to wear masks, eat outside and social distance from others on the trip. If they aren’t able to go, they say they will plan another trip in the U.S., potentially out West, though Ms. Cannon says they don’t have a solid plan B in mind yet. 

“If we are legally allowed to go, we will go,” she says. As of Monday afternoon, the couple said they hadn’t heard from their airline about any changes or restrictions.

Write to Allison Pohle at allison.pohle@wsj.com