The problem is the definition. Great Purge as itself wasn't a single event under Stalin's rule but waves of executions and convictions.
In fact after Lenin's death in 1922 Stalin came to power. With increasing intensity he started to deal with rivals, first politically, then he had enough power to order uncontrolled massacre. The most famous period is the interwar time period of 1937-1938 while he ordered millions of executions and forced labours. The problem is to define of the beginning since before 1937 and after 1938 there were numerous convictions and executions.
According to Russia Today (link)
Stalin's "Apparatus of Terror" relied mostly on the NKVD. Stalin's first purges date back to 1930–33 and were aimed at extermination of those who opposed industrialization and the kulaks (well-off farmers and entrepreneurs, who opposed collectivization).
His most ferocious acts of terror - The Great Purges - took place between 1934 and 1939
Actually Kirov was a rival of Stalin, with his death by murder (1st December 1934) the real terror in political level started. It was a good opportunity to Stalin to use it as excuse for bigger purge since Kirov was very (if not the most) popular party leader, so many people were outrageous about the event. Also Kirov was a party leader who had huge influence, with his disappearance Stalin's control over USSR greatly increased.
On the other end, 1939 wasn't the end of purges, for example Trotsky was murdered in August of 1940 in Mexico, but the intensity of purge decreased a lot under the war.