Abstract: The class number is an important invariant of an algebraic number field. In a sense, it is a measure of the arithmetic complexity of the field. Even in the seemingly simple case of quadratic fields, the most basic questions regarding the size of the class number, dating back to Gauss's Disquistiones Arithmeticae, remain unsolved. In this talk, after a brief description of the main objects of study, I will give an account of Yamamoto's work on a family of real quadratic fields with small class number.