  The toroidalization conjecture of D. Abramovich, K. Karu, K. Matsuki, and J.
Wlodarczyk asks whether any given morphism of nonsingular varieties over an
algebraically closed field of characteristic zero can be modified into a
toroidal morphism. Following a suggestion by Dale Cutkosky, we define the
notion of \emph{locally toroidal} morphisms and ask whether any locally
toroidal morphism can be modified into a toroidal morphism. In this paper, we
answer the question in the affirmative when the morphism is between any
arbitrary variety and a surface.
