  We explain various facets of the THSR (Tohsaki-Horiuchi-Schuck-R\"opke) wave
function. We first discuss the THSR wave function as a wave function of
cluster-gas state, since the THSR wave function was originally introduced to
elucidate the 3$\alpha$-condensate-like character of the Hoyle state ($0_2^+$
state) of $^{12}$C. We briefly review the cluster-model studies of the Hoyle
state in 1970's in order to explain how there emerged the idea to assign the
$\alpha$ condensate character to the Hoyle state. We then explain that the THSR
wave function can describe very well also non-gaslike ordinary cluster states
with spatial localization of clusters. This fact means that the dynamical
motion of clusters is of nonlocalized nature just as in gas-like states of
clusters and the localization of clusters is due to the inter-cluster Pauli
principle which is against the close approach of two clusters. The nonlocalized
cluster dynamics is formulated by the container model of cluster dynamics. The
container model describes gas-like state and non-gaslike states as the
solutions of the Hill-Wheeler equation with respect to the size parameter of
THSR wave function which is just the size parameter of the container. When we
notice that fact that the THSR wave function with the smallest value of size
parameter is equivalent to the shell-model wave function, we see that the
container model describes the evolution of cluster structure from the ground
state with shell-model structure up to the gas-like cluster state via ordinary
non-gaslike cluster states. For the description of various cluster structure,
more generation of THSR wave function have been introduced and we review some
typical examples with their actual applications.
