Our modern history of deep learning follows the arc of famous emergent disciplines in engineering (e.g. aero- and fluid dynamics) when theory lagged behind successful practical applications. Viewing neural networks from a dynamical systems perspective, in this work, we propose a novel characterization of deep neural networks as pointwise affine maps, making them accessible to a broader range of analysis methods to help close the gap between theory and practice. We begin by showing the equivalence of neural networks with parameter-varying affine maps parameterized by the state (feature) vector. As the paper's main results, we provide necessary and sufficient conditions for the global stability of generic deep feedforward neural networks. Further, we identify links between the spectral properties of layer-wise weight parametrizations, different activation functions, and their effect on the overall network's eigenvalue spectra. We analyze a range of neural networks with varying weight initializations, activation functions, bias terms, and depths. Our view of neural networks as affine parameter varying maps allows us to "crack open the black box" of global neural network dynamical behavior through visualization of stationary points, regions of attraction, state-space partitioning, eigenvalue spectra, and stability properties. Our analysis covers neural networks both as an end-to-end function and component-wise without simplifying assumptions or approximations. The methods we develop here provide tools to establish relationships between global neural dynamical properties and their constituent components which can aid in the principled design of neural networks for dynamics modeling and optimal control.