  In this paper we will introduce a certain type of morphisms of log schemes
(in the sense of Fontaine, Illusie, and Kato) and investigate their moduli.
Then by applying this we define a notion of toric algebraic stacks over
arbitrary schemes, which may be regarded as torus embeddings within the
framework of algebraic stacks, and study some basic properties. Furthermore, we
study the stack-theoretic analogue of toroidal embeddings.
