  This paper summarizes recent developments in the theory of
Bogomol'nyi-Prasad-Sommerfield (BPS) state counting and the wall crossing
phenomena, emphasizing in particular the role of the statistical mechanical
model of crystal melting. This paper is divided into two parts, which are
closely related to each other. In the first part, we discuss the statistical
mechanical model of crystal melting counting BPS states. Each of the BPS state
contributing to the BPS index is in one-to-one correspondence with a
configuration of a molten crystal, and the statistical partition function of
the melting crystal gives the BPS partition function. We also show that smooth
geometry of the Calabi-Yau manifold emerges in the thermodynamic limit of the
crystal. This suggests a remarkable interpretation that an atom in the crystal
is a discretization of the classical geometry, giving an important clue as to
the geometry at the Planck scale.In the second part we discuss the wall
crossing phenomena. Wall crossing phenomena states that the BPS index depends
on the value of the moduli of the Calabi-Yau manifold, and jumps along real
codimension one subspaces in the moduli space. We show that by using type
IIA/M-theory duality, we can provide a simple and an intuitive derivation of
the wall crossing phenomena, furthermore clarifying the connection with the
topological string theory. This derivation is consistent with another
derivation from the wall crossing formula, motivated by multi-centered BPS
extremal black holes. We also explain the representation of the wall crossing
phenomena in terms of crystal melting, and the generalization of the counting
problem and the wall crossing to the open BPS invariants.
