Abstract: The degree of mobility of a metric is the dimension of the space of solutions of a certain linear PDE system of finite type whose coefficients depend on the metric, and, for a given metric, there are standard algorithms to determine it. The standard algorithms strongly depend on the metric and in most cases it is possible to find the maximal and submaximal values of the degree of mobility only. I will show that the degree of mobility of a manifold is closely related to the space of parallel symmetric tensor fields on the cone over the manifold. In the case the metric is Einstein, it is essentially the tractor cone. I will use it to describe all possible values of the degree of mobility (on a simply connected manifold) for Riemannian and Lorenz metrics. . The part of results related to projective equivalence are joint with A. Fedorova and the part of results related to h-projective equivalence are joint with S. Rosemann.