Class numbers of quadratic fields have been the object of attention for many years, and there exist a large number of interesting results. This is a survey aimed at reviewing results concerning the divisibility of class numbers of both real and imaginary quadratic fields. More precisely, to review the question ‘do there exist infinitely many real (respectively imaginary) quadratic fields whose class numbers are divisible by a given integer?’ This survey also contains the current status of a quantitative version of this question.