In this thesis, we develop data-driven techniques to analyze unsteady aerodynamic flows under extremely gusty conditions for global field reconstruction, low-order modeling, and control. We first consider global field reconstruction from sparse sensors through the lens of generalized super-resolution analysis. This thesis offers a survey with comprehensive case studies of machine-learning-based super resolution for turbulent flows. Supervised machine-learning-based sparse reconstruction is then performed with vortical flows in a pump sump, an example of industrial turbulence. In addition, we establish a robust sparse reconstruction technique for situations in which the numbers and positions of sensors are changing over time, referred to as a Voronoi- tessellation-assisted convolutional neural network. We demonstrate its performance and robustness against noisy sensor measurements with a range of fluid flow examples. Defining interpolation and extrapolation conditions of machine-learning-based studies in unsteady flows is challenging due to their high-dimensionality and scale-invariant nature. For this reason, we consider nonlinear data-driven scaling of turbulent flows to reveal scale-invariant vortical structures across Reynolds numbers. This nonlinear scaling provides insights for supporting machine-learning-based studies of turbulent flows.To perform flow control leveraging the reconstructed fields from sparse sensors, we then consider constructing a control strategy of flows in a low-order subspace identified by nonlinear machine-learning-based data compression. We develop a nonlinear observable-augmented autoencoder that can incorporate physical observables in identifying a low-dimensional latent manifold. This thesis considers extreme vortex-gust airfoil interactions occurring when modern small aircraft fly in severe atmospheric conditions. Under such extreme aerodynamic situations, wings experience massive separation while exhibiting sharp and highly unsteady aerodynamic force responses. Although it is challenging to analyze the nonlinear, transient nature of extreme aerodynamics with conventional linear techniques, we reveal that the underlying physics of a collection of time-varying vortical flows in a high-dimensional space can be expressed on a low-rank manifold leveraging the present data-driven compression. It is also demonstrated that efficient control strategies can be derived at a minimal cost with the assistance of phase-amplitude reduction on the discovered manifold. These developed data-driven strategies offer a new perspective on reconstructing, modeling, and controlling a range of extremely unsteady flows.