Tui, which has a outlet in Westborough, Scarborough, and King Street, Bridlington, said all trips up to and including May 14 were being axed and its Marella Cruises sailings have been suspended until at least June.
It comes amid global travel restrictions due to COVID-19, which has infected more than 1.4 million people worldwide, with more than 82,000 deaths.
A Tui spokeswoman said: "We are constantly monitoring the situation and will start taking people on holiday again as soon as we are able to do so.
"At this point in time, nobody can accurately predict when that will be, so for the time being we will keep a close eye on our programme and continue to amend and adapt timings in line with the latest global travel advice."
Customers who have had their booking cancelled "will be able to amend their holiday to any other Tui package holiday on sale" on its website, she added.
The firm did not clarify whether consumers would have their bookings refunded.
Under EU law, travel companies must offer to refund customers within 14 days if their package holiday is cancelled but, facing many thousands of cancellations in recent weeks, the industry has warned that many firms will not survive if forced to do this immediately.
Trade association ABTA has called for the UK government to allow holiday companies to issue refund credit notes instead of cash refunds due to the deluge of claims caused by the coronavirus pandemic.