What is the Hellenistic culture?
Hellenistic culture is one of the most understudied and exciting areas of ancient history. Part of the reason for this is because it comes between the Greeks and the Romans (two favorites). Another reason is because Hellenistic culture is extremely expansive and difficult to study. After Phillip of Macedon unified the Greeks, there was a push towards Persia by Alexander, his son. Alexander got all the way to India. So, one can hellenisitc culture spans from Greece to India!
When Alexander died (323), his many leaders (diadochi) carved out his kingdom among themselves. For example, Ptolemy situated himself in Egypt and started the Ptolemaic empire. This was a fusion of Greek thought and Egyptian culture. Cassander took over Macedon. Seleucus was to take over the near east (parts of modern day Pakistan, Turkmenistan, etc.) To be honest, one must readily acknowledge the diversity within hellenism, but there were some commonalities. This can be called hellenistic culture. Some of these commanalities were: some sort of connection to the Greece through Alexander, Greek cultural influence, and Greek language.
For a brief book on hellenism, see: F. W. Walbank's Hellenistic World.
The Hellenistic culture is usually defined as the culture of the period from 336 BC to the time when the Roman Empire started to become very powerful in the late first century BCE.
The starting date for this period is chosen because that's when Phillip of Macedon, father of Alexander the Great, unified Macedonia (area north of the Greek peninsula) and the Greek city states.
During this period, particularly through the conquests of Alexander the Great, Greek culture spread far and wide, both into Asia and into Europe. For example, Greek culture was highly respected in the Roman Empire and the Greeks were seen as being more sophisticated and cultured than the Romans.
Although the Hellenistic Perod "officially" ended when Rome rose, the Hellenistic culture continued to be very influential.