This work deals with elliptical Wishart distributions on the set of symmetric positive definite matrices. It contains two major contributions. First, the information geometry associated with elliptical Wishart distributions is derived. Second, this geometry is leveraged to propose Riemannian-optimization-based maximum likelihood estimators of any elliptical Wishart distribution. Particular attention is given to two specific distributions: the t- and Kotz Wishart ones. The performance of the proposed methods is assessed through numerical experiments on simulated data.