The term Hellenistic (derived from Ἕλλην Héllēn, the  Greeks' traditional self-described ethnic name) was established by the  German historian Johann Gustav Droysen to refer to the spreading of  Greek culture over the non-Greek peoples that were conquered by  Alexander the Great. According to Droysen, the Hellenistic civilization  was a fusion of Greek and Near Eastern cultures. The main cultural  centers expanded from mainland Greece, to Pergamon, Rhodes, Antioch and  Alexandria.   
History
 
 Silver drachma of the Indo-Greek king Menander I (155-130 BC).
Obv: Greek legend, ΒΑΣΙΛΕΩΣ ΣΩΤΗΡΩΣ ΜΕΝΑΝΔΡΟΥ "[coin] of Saviour King Menander"
Rev: Kharosthi legend: MAHARAJA TRATASA MENADRASA "Saviour King Menander". Athena advancing right, with thunderbolt and shield. Taxila mint mark.
Source:Wikipedia
Obv: Greek legend, ΒΑΣΙΛΕΩΣ ΣΩΤΗΡΩΣ ΜΕΝΑΝΔΡΟΥ "[coin] of Saviour King Menander"
Rev: Kharosthi legend: MAHARAJA TRATASA MENADRASA "Saviour King Menander". Athena advancing right, with thunderbolt and shield. Taxila mint mark.
Source:Wikipedia
Modern historians see the death of Alexander  the Great in 323 BC as the beginning of the Hellenistic period.  Alexander's armies conquered the eastern Mediterranean, Egypt,  Mesopotamia, and the Iranian plateau, Central Asia, and parts of India.  Following Alexander's death, there was a struggle for the succession,  known as the wars of the Diadochi (Greek for successors).The struggle  ended in 281 BC with the establishment of four large territorial states.
- The Ptolemaic dynasty in Egypt based at Alexandria;
- The Seleucid dynasty in Syria and Mesopotamia based at Antioch;
- The Antigonid dynasty in Macedon and central Greece;
- The Attalid dynasty in Anatolia based at Pergamum.
His successors held on to the territory west of the  Tigris for some time and controlled the eastern Mediterranean until the  Roman Republic took control in the 2nd and 1st centuries BC. Most of the  east was eventually overrun by the Parthians, but Hellenistic culture  held on in distant locations, like the Greco-Bactrian kingdom in  Bactria, or the Indo-Greek kingdom in northern India, or the Cimmerian  Bosporus. Hellenistic culture remained dominant in the Eastern part of  the Roman Empire until its Christianisation and transition to the  Byzantine Empire.
Hellenism made considerable inroads  also in monarchies governed by kings of Persian, Armenian or Thracian  origin, as was the case with Armenia, Bithynia, Cappadocia and Pontus.
The end of the Hellenistic period is  generally seen as 31 BC, when the kingdom of Ptolemaic Egypt was utterly  defeated by the Romans at the Battle of Actium. As a result, Egypt's  last ruler, Cleopatra, committed suicide and her kingdom was annexed by  Octavian.
