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.