  In this paper we introduce a method which allows us to study properties of
the random uniform simplicial complex. That is, we assign equal probability to
all simplicial complexes with a given number of vertices and then consider
properties of a complex under this measure. We are able to determine or present
bounds for a number of topological and combinatorial properties. We also study
the random pure simplicial complex of dimension $d$, generated by letting any
subset of size $d+1$ of a set of $n$ vertices be a facet with probability $p$
and considering the simplicial complex generated by these facets. We compare
the behaviour of these models for suitable values of $d$ and $p$.
  Finally we use the equivalence between simplicial complexes and monotone
boolean functions to study the behaviour of typical such functions.
Specifically we prove that most monotone boolean functions are evasive, hence
proving that the well known Evasiveness conjecture is generically true for
monotone boolean functions without symmetry assumptions.
