Monaco does not have an international airport. The closest international airport is Nice airport.
Nice Airport Express runs bus 80 (formerly 110) from Terminals 1 and 2 to the Place d'Armes, Casino and Monte Carlo Bay Hotel and going on to the Menton bus station every hour or so starting at 8.15am and ending at 8.35pm. The ride takes about 45 minutes and costs €22 (33 round trip).
At night, there's the bus 601 (formerly 100N) which runs from 9:30pm to 2.30am and connects Parc Phoenix in Nice with Menton, Monaco, Beaulieu-sur-Mer and Roquebrune for €2.10.
Another possibility is to take the train from the Saint Augustin train station. To get to the St Augustin station follow the sign from outside Terminal 1. It's an easy 10-minute walk but if you're overloaded with luggage, take Tram 2 to the Grand Arenas stop.
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Trains between Nice and Monaco run about every half-hour until around 10pm, later on weekends. Try to avoid rush hours when it's sure to be standing room only and there are often delays. Friday afternoon trains returning to Nice are also jammed as shoppers from Ventimiglia's Friday market return with their haul.
The train station, Monaco/Monte Carlo is vast and well-equipped. Make sure to take the Port Hercule exit for the easiest access to Monaco Ville or Monte Carlo.
Check schedules and buy tickets on thetrainline.
Bus 600 route
Bus 600 links Monaco with Nice's Port Lympia. Bus 602 links Monaco with Nice's Vauban bus station. There is no bus station in Monaco. There are several stops as the bus traverses Monaco on its way to Menton: Place d'Armes, Train Station, Monte Carlo Casino. Services are handled by Zou.
See details about a day trip from Nice to Monaco
The most scenic approach to Monaco is via the Moyenne Corniche which winds along high above the coastline. There are even spots to safely pull over and take a photo. It's also a good way to avoid the traffic on the A8 motorway, especially during rush hour. On the Basse Corniche traffic can be heavy during the day.
A taxi from Nice to Monaco costs €60-€80 each way. It's somewhat cheaper to take an Uber but note that Uber is allowed to drop off passengers in Monaco but isn't allowed to pick up passengers in Monaco.