Due to romanization conventions, his name is sometimes written as "Khinchin" and other times as "Khintchine".

While studying at Moscow State University, he became one of the first followers of the famous Luzin school.

Khinchin graduated from the university in 1916 and six years later he became a full professor there, retaining that position until his death.

He became one of the founders of modern probability theory, discovering the law of the iterated logarithm in 1924, achieving important results in the field of limit theorems, giving a definition of a stationary process and laying a foundation for the theory of such processes.

Khinchin made significant contributions to the metric theory of Diophantine approximations and established an important result for simple real continued fractions, discovering a property of such numbers that leads to what is now known as Khinchin's constant.