We have two main purposes in this paper. One is to give some sufficient
conditions for the Julia set of a transcendental entire function $f$ to be
connected or to be disconnected as a subset of the complex plane ${\Bbb C}$.
The
other is to investigate the boundary of an unbounded periodic Fatou component
$U$, which is known to be simply-connected. These are related as follows: let
$\varphi : {\Bbb D} \longrightarrow U$ be a Riemann map of $U$ from a unit
disk
${\Bbb D}$, then under some mild conditions we show that the set
$\Theta_{\infty}$
of all angles where $\varphi$ admits the radial limit $\infty$ is dense in
$\partial {\Bbb D}$ if $U$ is an attracting basin, a parabolic basin or a
Siegel
disk. If $U$ is a Baker domain on which $f$ is not univalent, then
$\Theta_{\infty}$ is dense in $\partial {\Bbb D}$ or at least its closure
$\overline{\Theta_{\infty}}$ contains a certain perfect set, which means the
boundary $\partial U$ has a very complicated structure. In all cases, this
result leads to the disconnectivity of the Julia set $J_f$ in ${\Bbb C}$. If
$U$ is
a Baker domain on which $f$ is univalent, however, we shall show by giving an
example that $\partial U$ can be a Jordan arc in ${\Bbb C}$, which has a
rather
simple structure, and, moreover, $J_f$ can be connected.

We also consider the connectivity of the set $J_f \cup \{ \infty \}$ in the
Riemann sphere $\widehat{{\Bbb C}}$ and show that $J_f \cup \{ \infty \}$ is
connected if
and
only if $f$ has no multiply-connected wandering domains.