  We introduce a fermionic formula associated with any quantum affine algebra
U_q(X^{(r)}_N). Guided by the interplay between corner transfer matrix and
Bethe ansatz in solvable lattice models, we study several aspects related to
representation theory, most crucially, the crystal basis theory. They include
one dimensional sums over both finite and semi-infinite paths, spinon character
formulae, Lepowski-Primc type conjectural formula for vacuum string functions,
dilogarithm identities, Q-systems and their solution by characters of various
classical subalgebras and so forth. The results expand [HKOTY1] including the
twisted cases and more details on inhomogeneous paths consisting of non-perfect
crystals. As a most intriguing example, certain inhomogeneous one dimensional
sums conjecturally give rise to branching functions of an integrable
G^{(1)}_2-module related to the embedding G^{(1)}_2 \hookrightarrow B^{(1)}_3
\hookrightarrow D^{(1)}_4.
