Is there a topological characterisation of what a (closed irreducible) hyperbolic 3manifold is? I don't know any Riemannian geometry and still want to understand what an exceptional Dehn surgery is. For definition of a hyperbolic knot we can avoid the real understanding of what a hyperbolic manifold is by simply saying that it is not an unknot, not a torus knot and not a satellite. Is there a similar characterisation for hyperbolic 3manifolds? Cameron Gordon's Park City lecture notes seem to suggest that such a criterion would be that the manifold is simple. But I couldn't find a clear statement of this. If there is such a criterion, a reference for a survey discussing these things would be very desirable(and in general, is there a survey  suitable for early stage PhD students  about the 3manifold topology after Perelman?). Thanks!

A clear statement is the following:
You may remember that by saying that $M$ is hyperbolic unless there is some clear obstruction, and the obstrucion is $\pi_1$ finite or the existence of some surface with $\chi \geqslant 0$. This statement is pretty general and works for any compact 3manifold with any kind of (possibly empty) boundary, using the appropriate notions of "hyperbolic" and "essential".
More concretely, an orientable manifold $M$ is hyperbolic unless one of the following obstructions occurs:
This characterization implies the one for knots that you mentioned, in virtue of the following:


