Hurricane Delta has made landfall near Puerto Morelos on Mexico's Caribbean coast.
The hurricane has weakened to a Category 2 storm with winds of 175km/h (110mph).
The US National Hurricane Center has warned of life-threatening storm surges and strong winds.
It said the weather was deteriorating quickly over the north-eastern Yucatán coast, an area popular with tourists from Mexico and abroad.
Thousands of tourists and residents have moved into shelters for safety.
Delta is the 26th named storm of the Atlantic hurricane season, which typically runs from June to November, and has had an unusually high number of storms this year.
A hurricane warning is in effect between Dzilam and Tulum, including Cozumel.
Torrential rains have been forecast for the states of Campeche, Quintana Roo and Yucatán.
The National Hurricane Center says the storm surge could raise water levels along the northern part of the Yucatán Peninsula by between 2.4m and 3.7m (8-12ft). It says large and destructive waves are expected along the coast.
More than 150 shelters have opened their doors to people seeking refuge from the storm in the resort of Cancún.
The head of the area's hotel association, Roberto Citrón, told AFP news agency that "to prevent the spread of Covid-19, the same measures have been taken in the shelters as in hotels, such as the use of gel and face masks".
Oil companies operating in the Gulf of Mexico have also told non-essential staff to evacuate their offshore platforms and Mexican ports have been closed along its eastern coast.