In paleolithic, many 40s-10s of thousand of years ago, prehistory area of the city was thick forests with small settlements along the rivers. Romania as a whole though has long human habitation, the second oldest European human remains (that were hybrid Neanderthal) were found in NE Romania were about 40,000 years old. An ancient culture existed here that became the people group here and all around Romania. Tribes that were collectively described as Getae by Ancient Greeks (people north of the Danube River). Romans later observed them calling eachother Daci so name stuck, the Dacians. A settlement on river near site of Radu Vadu Monastery was found through archaeological evidence to be inhabited at least temporarily many tens of thousands of years, potentially over 50,000 years! Incredible. A permanent settlements evidence (unofficial founding of a city here) was there since 1800BC. 6 different Dacian settlements/villages, have been unearthed inside current boundaries of the metro area. They had trade with Greeks and Romans as their coins and jewelry were found in excavations. 

Dacian tribes inhabited all of present day Romania, Moldova, and parts of Ukraine, North Bulgaria, Serbia, Hungary, Slovakia, and Southern Poland. They spoke a language related to Thracian, from south of here in Bulgaria. They were influenced by nomadic Scythians who in 9000BC moved from their origins in Iran to Caucuses and then areas of Ukraine and Southern Russia, then to here by 700BC assimilating into the population. Also in 500BC Celtic invasions influenced the tribes also. In 440BC the Dacians handed defeats to the great Persian Empire in their attempt to conquer the Scythians (I guess the Persian Army didn’t realize they’d run into Dacians also). And 350s Alexander the Great attacked as well. 

Dacian culture and religion worshipped priest kings and believed people were immortal, death was a change of country of sorts, for the soul. They worshipped the sun and spirit world. Zalmoxis was main God of thunder and lightning, and others like God of the moon, hunting, and human health were other deities. Some of their cults were even spread into ancient Greece, one being so popular it became an official holiday festival in Athens around 400BC. Greek historian Strabo wrote of a Dacian priest king travelling south all the way to Egypt on a religious spiritual journey in around 500BC, learning about every culture and religion along the way. The Dacian religion was still practiced long into Christian times until about 500AD actually before they were Christianized.

A powerful Dacian Kingdom was formed in 168 BC that ended up having near the same boundaries of modern Romania that lasted until 106AD with capital in present Transylvania. A powerful king Burebista, a contemporary of Julius Caesar, expanded the territory and expelled Celts who been living there a few hundred years, causing Caesar to formulate plans to conquer the threat. Romans had suffered defeats from Dacia prior to Caesar in some dustups along the boundary. King Burebista was killed internally though and Dacia was split in 4 parts in 44BC weakening the kingdom.
 One even made alliances with Rome in intermarrying the Roman Emperors daughters. There was still a sizable Dacian Kingdom though with Roman areas on its periphery, which continued to pose a threat to Rome and also was tempting because of vast riches of gold. 85AD the Dacians were reunited once again.80sAD there were many battles with Dacians and Romans which went both ways but by 105 AD, Emperor Trajan legions conquered capital in Transylvania and much of modern Romania (but not all, like this area of Bucharest fought them off and remained independent). The free Dacians. Emperor Trajan has the largest ever Roman victory celebration financed by Dacian gold and carved the victory into famous Trajan's column in Rome. 

The Province of Roman Dacia lasted until 271. Romans intermixed with the Dacians and this is where the origins of Romanian culture and language come from. Not slav or other migrations of people like all of their neighbors. Thus the name (ROM(E)ania. Goths and Carpis raiders came in from the north and east respectively in 271 took over area. Roman aristocracy fled and it was first ever Roman province to be abandoned.  I will get more into Roman history on a later date but since they never conquered area of city except briefly in 330s by Constantine legions, I will skip details on Roman Dacia. Area around Bucharest and South and East Romania remained independent Dacian chiefdoms.
AND REST OF HISTORY IN COMMENTS because it'd didnt fit