This article investigates the parameter space of the exponential family $z\mapsto \exp(z)+\kappa$. We prove that the boundary (in $\mathbb{C}$) of every hyperbolic component is a Jordan arc, as conjectured by Eremenko and Lyubich as well as Baker and Rippon, and that $\infty$ is not accessible through any nonhyperbolic ("queer") stable component. The main part of the argument consists of demonstrating a general "Squeezing Lemma", which controls the structure of parameter space near infinity. We also prove a second conjecture of Eremenko and Lyubich concerning bifurcation trees of hyperbolic components.