It is quite easy to compute the continued fraction of an irrational number up to any prescribed length. Usually, such expansions show no pattern; Euler’s number e is an exception. In this chapter we investigate other exceptions, namely, quadratic irrationals (that are algebraic numbers of degree two). Their continued fractions have many patterns which imply interesting properties with respect to diophantine approximations. However, this chapter is not only about quadratic irrationals; it also provides a classification of all real numbers with respect to their continued fraction expansion and aspects of diophantine approximations.