  We study 1-way quantum finite automata (QFAs). First, we compare them with
their classical counterparts. We show that, if an automaton is required to give
the correct answer with a large probability (over 0.98), then the power of
1-way QFAs is equal to the power of 1-way reversible automata. However, quantum
automata giving the correct answer with smaller probabilities are more powerful
than reversible automata.
  Second, we show that 1-way QFAs can be very space-efficient. Namely, we
construct a 1-way QFA which is exponentially smaller than any equivalent
classical (even randomized) finite automaton. This construction may be useful
for design of other space-efficient quantum algorithms.
  Third, we consider several generalizations of 1-way QFAs. Here, our goal is
to find a model which is more powerful than 1-way QFAs keeping the quantum part
as simple as possible.
