While a student of Andrey Kolmogorov at Moscow State University and still a teenager, Arnold showed in 1957 that any continuous function of several variables can be constructed with a finite number of two - variable functions, thereby solving Hilbert's thirteenth problem.

After graduating from Moscow State University in 1959, he worked there until 1986 (a professor since 1965), and then at Steklov Mathematical Institute.

He became an academician of the USSR Academy of Sciences (Russian Academy of Science since 1991) in 1990. Arnold can be said to have initiated the theory of symplectic topology as a distinct discipline. The Arnold conjecture on the number of fixed points of Hamiltonian symplectomorphisms and Lagrangian intersections were also a major motivation in the development of Floer homology.

Arnold worked at the Steklov Mathematical Institute in Moscow and at Paris Dauphine University up until his death. As of 2006 he was reported to have the highest citation index among Russian scientists, and h-index of 40.

To his students and colleagues Arnold was known also for his sense of humour. For example, once at his seminar in Moscow, at the beginning of the school year, when he usually was formulating new problems, he said:

There is a general principle that a stupid man can ask such questions to which one hundred wise men would not be able to answer. In accordance with this principle I shall formulate some problems.

Arnold died of peritonitis on 3 June 2010 in Paris, nine days before his 73rd birthday. His students include Alexander Givental, Victor Vassiliev and Askold Khovanskii.

Arnold is well known for his lucid writing style, combining mathematical rigour with physical intuition, and an easy conversational style of teaching. His writings present a fresh, often geometric approach to traditional mathematical topics like ordinary differential equations, and his many textbooks have proved influential in the development of new areas of mathematics. However, Arnold's books have been criticized for supporting the theory with statements meant to teach an intuitive understanding, without providing the tools necessary to prove these statements.

Arnold was an outspoken critic of the trend of high levels of abstraction in mathematics during the middle of the last century. He had very strong opinions on how this approach — which was most popularly implemented by the Bourbaki school in France — initially had a negative impact on French, and then later other countries', mathematical education. Arnold's sharp critique of Bourbaki approach was in turn criticized by other mathematicians.