The Event Horizon Telescope (EHT) collaboration has recently released the first image of a black hole (BH), opening a new window onto tests of general relativity in the strong field regime. In this paper, we derive constraints on the nature of M87* (the supermassive object at the center of the galaxy M87), exploiting the fact that its shadow appears to be highly circular, and using measurements of its angular size. We first consider the simple case where M87* is assumed to be a Kerr BH. We find that the inferred circularity of M87* excludes Kerr BHs with observation angle theta(obs) greater than or similar to 45 degrees for dimensionless rotational parameter 0.95 less than or similar to a(*) <= 1 whereas the observation angle is unbounded for a(*) less than or similar to 0.9. We then consider the possibility that M87* might be a superspinar, i.e., an object described by the Kerr solution and spinning so fast that it violates the Kerr bound by having vertical bar a(*)vertical bar > 1. We find that, within certain regions of parameter space, the inferred circularity and size of the shadow of M87* do not exclude the possibility that this object might be a superspinar.