  We construct an entire function in the Eremenko-Lyubich class $\B$ whose
Julia set has only bounded path-components. This answers a question of Eremenko
from 1989 in the negative.
  On the other hand, we show that for many functions in $\B$, in particular
those of finite order, every escaping point can be connected to $\infty$ by a
curve of escaping points. This gives a partial positive answer to the
aforementioned question of Eremenko, and answers a question of Fatou from 1926.
