One way to define random graphs is to fix a finite set of properties $P$. Picking with equal probability a graph with $|V|=n$ out of the set $\Gamma_n(P)$ of all $n$-(vertex-)graphs with properties $P$ yields a random $n$-$P$-graph. This is the theoretical point of view. Practically, the set is not at hand and one has to generate with equal probability graphs that happen to be in $\Gamma_n(P)$.
The classical Erdös-Renyi graphs $G(n,m)$ are defined like this with $P = \{|E| = m\}$. To generate an Erdös-Renyi random graph is easy.
Configuration models allow to generate random graphs with a given degree distribution. Advanced configuration models allow to generate graphs that in addition have a given distribution of the clustering coefficient.
My questions are:
For which classes of random graphs (defined by a set of properties they must have) are there well known and efficient ways to generate them randomly? (This might be a big list question.)
For which classes of random graphs are no ways known to efficiently generate them randomly?
By which heuristic arguments can one tell whether for a given set of (non-contradictory) properties there may be ways to efficiently generate them randomly - or not?