  This paper studies Random and Evolving Apollonian networks (RANs and EANs),
in d dimension for any d>=2, i.e. dynamically evolving random d dimensional
simplices looked as graphs inside an initial d-dimensional simplex. We
determine the limiting degree distribution in RANs and show that it follows a
power law tail with exponent tau=(2d-1)/(d-1). We further show that the degree
distribution in EANs converges to the same degree distribution if the
simplex-occupation parameter in the n-th step of the dynamics is q_n->0 and
sum_{n=0}^infty q_n =infty. This result gives a rigorous proof for the
conjecture of Zhang et al. that EANs tend to show similar behavior as RANs once
the occupation parameter q->0. We also determine the asymptotic behavior of
shortest paths in RANs and EANs for arbitrary d dimensions. For RANs we show
that the shortest path between two uniformly chosen vertices (typical
distance), the flooding time of a uniformly picked vertex and the diameter of
the graph after n steps all scale as constant times log n. We determine the
constants for all three cases and prove a central limit theorem for the typical
distances. We prove a similar CLT for typical distances in EANs.
