  In this paper, we introduce and explore a new model of {\it quantum finite
automata} (QFA). Namely, {\it one-way finite automata with quantum and
classical states} (1QCFA), a one way version of {\it two-way finite automata
with quantum and classical states} (2QCFA) introduced by Ambainis and Watrous
in 2002 \cite{AJ}. First, we prove that {\it one-way probabilistic finite
automata} (1PFA) \cite{AP} and {\it one-way quantum finite automata with
control language} (1QFACL) \cite{ACB} as well as several other models of QFA,
can be simulated by 1QCFA. Afterwards, we explore several closure properties
for the family of languages accepted by 1QCFA. Finally, the state complexity of
1QCFA is explored and the main succinctness result is derived. Namely, for any
prime $m$ and any $\epsilon_1>0$, there exists a language $L_{m}$ that cannot
be recognized by any {\it measure-many one-way quantum finite automata}
(MM-1QFA) \cite{Kon97} with bounded error $7/9+\epsilon_1$, and any 1PFA
recognizing it has at last $m$ states, but $L_{m}$ can be recognized by a 1QCFA
for any error bound $\epsilon>0$ with $\bf{O}(\log{m})$ quantum states and 12
classical states.
