  We analyze the notion of guessing model, a way to assign combinatorial
properties to arbitrary regular cardinals. Guessing models can be used, in
combination with inaccessibility, to characterize various large cardinals
axioms, ranging from supercompactness to rank-to-rank embeddings. The majority
of these large cardinals properties can be defined in terms of suitable
elementary embeddings j\colon V_\gamma \to V_\lambda. One key observation is
that such embeddings are uniquely determined by the image structures j [
V_\gamma ]\prec V_\lambda. These structures will be the prototypes guessing
models. We shall show, using guessing models M, how to prove for the ordinal
\kappa_M=j_M (\crit(j_M)) (where \pi_M is the transitive collapse of M and j_M
is its inverse) many of the combinatorial properties that we can prove for the
cardinal j(\crit(j)) using the structure j[V_\gamma]\prec V_{j(\gamma)}.
\kappa_M will always be a regular cardinal, but consistently can be a
successor. Guessing models M with \kappa_M=\aleph_2 exist assuming the proper
forcing axiom PFA. By means of these models we shall introduce a new structural
property of models of PFA: the existence of a "Laver function" f : \aleph_2 \to
H_{\aleph_2} sharing the same features of the usual Laver functions f
:\kappa\to H_\kappa provided by a supercompact cardinal \kappa. Further
applications of our analysis will be proofs of the singular cardinal hypothesis
and of the failure of the square principle assuming the existence of guessing
models. In particular the failure of square shows that the existence of
guessing models is a very strong assumption in terms of large cardinal
strength.
