(CNN) Brussels' transport system is largely on lockdown and foreign governments have warned their citizens to be vigilant when visiting the country, following two deadly explosions at the city's airport and one at Maelbeek metro station on Tuesday.

Flights were diverted away from the airport following the attacks, according to the country's public broadcaster, RTBF.

IMAGE: Flights being diverted away from Brussels airport pic.twitter.com/AXgn9IuoTk

Brussels Airport is currently closed and its CEO Arnaud Feist has confirmed that it will remain closed through Wednesday while officials evaluate when to resume operations.

"Brussels Airport has been shot in the heart," he said. "We'd like to thank the police and rescue services and the people present at the terminal."

Hello from Brussels. Following the attacks, there will be no flights into and out of #brusselsairport today.

"As soon as we have access to the terminal building, we can assess the damage," the airport said Wednesday on Twitter.

"Later today we will assess when operations can be resumed."

The entire Brussels public transport system was closed early Tuesday, but select metro, tram and bus lines have resumed service, according to the system's web site

European rail provider Eurostar suspended service to Brussels on Tuesday and advised passengers to postpone travel and avoid Brussels Midi Station.

Eurostar expects to resume regular service on Wednesday, the rail provider said on Twitter. Travelers are advised to arrive on time and allot at least an hour for check in.

We'll run a normal service on 23/03 if you prefer not to travel to/from Brussels you can exchange free of charge https://t.co/hkZ0jEUaIj — Eurostar (@Eurostar) March 22, 2016

Paris' Gare du Nord station -- from which Eurostar services leave for London -- was evacuated on Tuesday afternoon following the discovery of an abandoned suitcase.

The station has since reopened and police in Paris have confirmed to CNN that the security alert was a false alarm.

Around the world

Outside of Belgium, security has tightened at European airports, and U.S. cities and airports have also ramped up security

The U.S. State Department issued an alert to U.S. citizens traveling to and throughout Europe, noting that terrorist groups continue to plan near-term attacks throughout Europe, targeting sporting events, tourist sites, restaurants, and transportation.

"U.S. citizens should exercise vigilance when in public places or using mass transportation," said the alert , issued March 22.

"Be aware of immediate surroundings and avoid crowded places. Exercise particular caution during religious holidays and at large festivals or events."

The British government has also warned its citizens to be vigilant when visiting Belgium, which has now been placed at threat level 4 -- meaning "a serious and imminent threat."

"You should remain alert and vigilant, especially in places where there's a high concentration of people," said the UK Home Office.

"Belgian security operations are likely to be carried out at short notice. If you're in an affected area you should follow the instructions of the Belgian security authorities. Police have asked the public not to comment on police operations on social media," it added.

Similarly, the Australian government has urged citizens to use a "high degree of caution" when traveling to Brussels.

"Travelers are reminded that there is a heightened potential for police raids to take place with little or no warning in response to the raised terrorism threat," it said

The Netherlands is advising people not to travel to Brussels, according to new travel advice for Belgium on the Foreign Ministry's website

Dutch Foreign Minister Bert Koenders urges the public to refrain from traveling to Brussels until more is known about the situation.

"Those who are in Brussels already are advised to stay indoors wherever possible," said the alert. "The city's metro stations and airport should also be avoided. It is important that the public follow the instructions given by the authorities.'

Meanwhile Japan's Ministry of Foreign Affairs is advising all Japanese travelers overseas to inform their whereabouts to their families, friends or employers.

Travelers who plan on staying abroad for more than three months are also obligated to submit an "Overseas Residential Registration" to the Ministry of Foreign Affairs.

The ministry urges Japanese travelers currently traveling to or staying in Belgium to exercise special caution at all times.

Similar warnings have been issued by officials in other countries, including Canada, New Zealand, the Philippines, Hong Kong SAR and India.

Japanese automobile manufacturers Toyota Motor Corporation and Nissan Motor Company Ltd have prohibited their employees from traveling to Belgium following the attacks.

Panasonic told CNN it is still confirming the whereabouts of its employees and their families, and hasn't released a travel ban to Belgium yet.

Canceled flights

Lufthansa, Air France, KLM, British Airways, American Airlines, All Nippon Airways, United Airlines, Delta Airlines, Ryanair, Alitalia, Jet Airways and Easyjet have all canceled or diverted flights.

Aegean is offering free rebookings or full refunds to any passengers who want to cancel their Brussels flights between now and March 31.

Aer Lingus is offering free changes or refunds to customers traveling on affected Brussels flights on March 22 and 23.

Air Malta is waiving re-booking fees for customers with flights to and from Brussels Airport, up to March 24.

Many other airlines are offering similar options to affected travelers.