  Random planar graphs have been the subject of much recent work. Many basic
properties of the standard uniform random planar graph P_{n}, by which we mean
a graph chosen uniformly at random from the set of all planar graphs with
vertex set {1,2,...,n}, are now known, and variations on this standard random
graph are also attracting interest.
  Prominent among the work on P_{n} have been asymptotic results for the
probability that P_{n} will be connected or contain given components/subgraphs.
Such progress has been achieved through a combination of counting arguments and
a generating function approach.
  More recently, attention has turned to P_{n,m}, the graph taken uniformly at
random from the set of all planar graphs on {1,2,...,n} with exactly m(n) edges
(this can be thought of as a uniform random planar graph with a constraint on
the average degree). The case when m(n) = qn for fixed q in (1,3) has been
investigated, and results obtained for the events that P_{n,qn} will be
connected and that P_{n,qn} will contain given subgraphs.
  In Part I of this thesis, we use elementary counting arguments to extend the
current knowledge of P_{n,m}. We investigate the probability that P_{n,m} will
contain given components, the probability that P_{n,m} will contain given
subgraphs, and the probability that P_{n,m} will be connected, all for general
m(n), and show that there is different behaviour depending on which `region'
the ratio m(n)/n falls into. In Part II, we investigate the same three topics
for a uniform random planar graph with constraints on the maximum and minimum
degrees.
