Abstract: Gaussian mixture models are widely used in Statistics. A fundamental aspect of these distributions is the study of the local maxima of the density function, or modes. In particular, it is not known how many modes a mixture of k Gaussians in d dimensions can have. In this talk I will give a brief account of this problem's history, present lower bounds and the first upper bound on the maximum number of modes, provided it is finite. This is joint work with Christian Haase and Alexander Engström.