Many of the deepest results in polynomial dynamics are obtained using dynamic rays and their landing properties. The rays are constructed using the simple form of the dynamics near infinity. For transcendental entire functions, infinity is an essential singularity without simple normal form. Fatou and Eremenko asked whether points converging to infinity (”escaping points”) have the form of curves to infinity; escaping points are always contained in the Julia set. We show that this true for large classes of bounded type entire functions, including those of finite order. We also show that there is a bounded type entire function for which every path component of the Julia set, and especially of the set of escaping points, is bounded.