  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 centre 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 $\theta_{\rm obs} \gtrsim 45^{\circ}$
for dimensionless rotational parameter $0.95 \lesssim a_* \leq 1$ whereas the
observation angle is unbounded for $a_* \lesssim 0.9$. 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
$|a_*| > 1$. 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.
