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The public transit system, run by Société de transport de Montréal (STM) , is safe, efficient, and pleasant to use. Tickets have been replaced by cards with magnetic stripe containing one trip, called an à la carte ticket. These are valid for one trip (including transfer) on the metro and buses cost $2.75 each, but are also available for about 22.75% less when you purchase six for $12.75 either from the Metro agent or the automatic fare vending machine located in Metro stations.
You need to keep your payment card as it is your proof of payment and transfer (correspondance).
If you are using cash to pay your fare on the bus, it is important to have the exact fare since the driver does not give change; you will receive an à la carte ticket, this is your proof of payment and your transfer.
Pictures and specific instructions can be found here: 
Tourist passes offer unlimited travel on the bus and metro for periods of one day ($9) or three days ($17) and are well worth it. They are available from most downtown metro stations during the summer, but only at Berri-UQAM, Peel, and Bonaventure stations on the off-season. Weekly ($20.00 regular, $11.25 for students under 18; valid from the nearest Sunday of purchase) and monthly ($68.50 regular, $37 for students under 25) passes are also available. Only students studying at a recognized academic institution in Montréal may benefit from student fares and a special card must be obtained from the STM.
The STM has stopped issuing disposable monthly passes and has started using the reusable OPUS card. The OPUS card is a smart card with a chip that contains your fare and transfer information. The OPUS card can be purchased at all metro stations and transit fare points of sale. As of December 2009 the card costs $3.50 
OPUS cards can be refilled at Metro stations using the automated machines or at the ticket booth.
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