You’ll find hundreds of bank branches and ATMs in the centre. Most banks have a currency-exchange office and can provide cash advances against credit or debit cards. Always bring your passport, since you will likely have to show it. Most banks operate only on weekdays, though some may have limited Saturday morning hours.
Outside of normal banking hours, you can change money at private currency booths (casa de schimb). There is a row of these along B-dul Gen Gheorghe Magheru, running north of Piaţa Universităţii. We generally don’t recommend using these, as the rates they offer tend to be the same as or lower than the banks', often with higher commission fees. If you do exchange money at a private booth, before you surrender your cash, tell the cashier exactly what you want to exchange and ask him or her to write down the amount you will receive in lei. You'll usually have to show a passport here as well.