We have two main purposes in this paper. One is to give some sufficientconditions for the Julia set of a transcendental entire function $f$ to beconnected or to be disconnected as a subset of the complex plane ${\Bbb C}$.Theother 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 unitdisk${\Bbb D}$, 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 aSiegeldisk. 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 theboundary $\partial U$ has a very complicated structure. In all cases, thisresult leads to the disconnectivity of the Julia set $J_f$ in ${\Bbb C}$. If$U$ isa Baker domain on which $f$ is univalent, however, we shall show by giving anexample that $\partial U$ can be a Jordan arc in ${\Bbb C}$, which has arathersimple structure, and, moreover, $J_f$ can be connected.

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