  A complete deterministic finite (semi)automaton (DFA) with a set of states
$Q$ is \emph{completely reachable} if every nonempty subset of $Q$ is the image
of the action of some word applied to $Q$. The concept of completely reachable
automata appeared, in particular, in connection with synchronizing automata;
the class contains the \v{C}ern{\'y} automata and covers several independently
investigated subclasses. The notion was introduced by Bondar and Volkov (2016),
who also raised the question about the complexity of deciding if an automaton
is completely reachable. We develop an algorithm solving this problem, which
works in ${\O(|\Sigma|\cdot n^2)}$ time and $\O(|\Sigma|\cdot n)$ space, where
$n=|Q|$ is the number of states and $|\Sigma|$ is the size of the input
alphabet. In the second part, we prove a weak Don's conjecture for this class
of automata: a subset of states $S \subseteq Q$ is reachable with a word of
length at most $2n(n-|S|) - n \cdot H_{n-|S|}$, where $H_i$ is the $i$-th
harmonic number. This implies a quadratic upper bound in $n$ on the length of
the shortest synchronizing words (reset threshold) for the class of completely
reachable automata and generalizes earlier upper bounds derived for its
subclasses.
