  This is an extended version of the thesis presented to the Programa de
P\'os-Gradua\c{c}\~ao em Matem\'atica of the Departamento de Matem\'atica,
PUC-Rio, in September 2013, incorporating some suggestions from the examining
commission.
  Random graphs (and more generally hypergraphs) have been extensively studied,
including their first order logic. In this work we focus on certain specific
aspects of this vast theory. We consider the binomial model $G^{d+1}(n,p)$ of
the random $(d+1)$-uniform hypergraph on $n$ vertices, where each edge is
present, independently of one another, with probability $p=p(n)$. We are
particularly interested in the range $p(n) \sim C\log(n)/n^d$, after the double
jump and near connectivity. We prove several zero-one, and, more generally,
convergence results and obtain combinatorial applications of some
