  We study the computational complexity of the problem $\#\text{IndSub}(\Phi)$
of counting $k$-vertex induced subgraphs of a graph $G$ that satisfy a graph
property $\Phi$. Our main result establishes an exhaustive and explicit
classification for all hereditary properties, including tight conditional lower
bounds under the Exponential Time Hypothesis (ETH):
  - If a hereditary property $\Phi$ is true for all graphs, or if it is true
only for finitely many graphs, then $\#\text{IndSub}(\Phi)$ is solvable in
polynomial time.
  - Otherwise, $\#\text{IndSub}(\Phi)$ is $\#\mathsf{W[1]}$-complete when
parameterised by $k$, and, assuming ETH, it cannot be solved in time $f(k)\cdot
|G|^{o(k)}$ for any function $f$.
  This classification features a wide range of properties for which the
corresponding detection problem (as classified by Khot and Raman [TCS 02]) is
tractable but counting is hard. Moreover, even for properties which are already
intractable in their decision version, our results yield significantly stronger
lower bounds for the counting problem. As additional result, we also present an
exhaustive and explicit parameterised complexity classification for all
properties that are invariant under homomorphic equivalence. By covering one of
the most natural and general notions of closure, namely, closure under
vertex-deletion (hereditary), we generalise some of the earlier results on this
problem. For instance, our results fully subsume and strengthen the existing
classification of $\#\text{IndSub}(\Phi)$ for monotone (subgraph-closed)
properties due to Roth, Schmitt, and Wellnitz [FOCS 20].
