  In this letter, we propose a simple rule that generates scale-free networks
with very large clustering coefficient and very small average distance. These
networks are called {\bf Random Apollonian Networks}(RANs) as they can be
considered as a variation of Apollonian networks. We obtain the analytic result
of power-law exponent $\gamma =3$ and clustering coefficient
$C={46/3}-36\texttt{ln}{3/2}\approx 0.74$, which agree very well with the
simulation results. We prove that the increasing tendency of average distance
of RAN is a little slower than the logarithm of the number of nodes in RAN.
Since many real-life networks are both scale-free and small-world, RANs may
perform well in mimicking the reality. The epidemic spreading process is also
studied, we find that the diseases spread slower in RANs than BA networks in
the early stage of SI process, indicating that the large clustering coefficient
may slower the spreading velocity especially in the outbreaks.
