A Hankel matrix is an infinite matrix of the form {a(n+m)}, where n, m are non-negative integers. A multiplicative Hankel matrix is an infinite matrix of the form {a(nm)} (the argument of a is the product of n and m), where n and m are positive integers. The theory of Hankel matrices is classical and well established, while the theory of their multiplicative analogues seems to be in its infancy. I will attempt to give a survey and comparison of these two theories; topics to be covered are: boundedness, positive definiteness, finite rank properties, and spectral analysis of some concrete examples.