  This survey studies asymptotics of random fringe trees and extended fringe
trees in random trees that can be constructed as family trees of a
Crump-Mode-Jagers branching process, stopped at a suitable time. This includes
random recursive trees, preferential attachment trees, fragmentation trees,
binary search trees and (more generally) $m$-ary search trees, as well as some
other classes of random trees.
  We begin with general results, mainly due to Aldous (1991) and Jagers and
Nerman (1984). The general results are applied to fringe trees and extended
fringe trees for several particular types of random trees, where the theory is
developed in detail. In particular, we consider fringe trees of $m$-ary search
trees in detail; this seems to be new.
  Various applications are given, including degree distribution, protected
nodes and maximal clades for various types of random trees. Again, we emphasise
results for $m$-ary search trees, and give for example new results on protected
nodes in $m$-ary search trees.
  A separate section surveys results on height, saturation level, typical depth
and total path length, due to Devroye (1986), Biggins (1995, 1997) and others.
  This survey contains well-known basic results together with some additional
general results as well as many new examples and applications for various
classes of random trees.
