It is easy and cheap to travel around Montevideo on buses, which run all over the city until around 11pm daily. Tickets are bought directly from the driver or conductor, who can also advise on routes and destinations. Locals are friendly and willing to help advise on bus routes to access the various sights and attractions; a Spanish phrase book is useful for this purpose. Taxis are plentiful in the city. They are metered and can be expensive, particularly if you are transporting luggage (there is a surcharge).
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