The past twenty years have witnessed the advent and extensive study of metamaterials and, more recently, metasurfaces. These concepts have widely extended the range of available material properties and have enabled new or enhanced wave-propagation effects, from negative refraction and invisibility to light trapping and antenna beam shaping. In this dissertation, we use, or take inspiration from, metamaterials and metasurfaces to engineer the electromagnetic scattering and the wavefront of propagating waves for various relevant applications. We note that the “optical theorem” of scattering theory relates the concepts of scattering engineering and wavefront manipulation and provides a general framework for many of the ideas discussed in this dissertation. While this work is mostly theoretical and computational, considering the potential experimental demonstration of the proposed ideas, we focused on realistic material platforms and structures that can be fabricated with state-of-the-art technologies. The first part of this dissertation is devoted to one of the most important scattering-engineering problems, namely, scattering/reflection reduction and invisibility. To demonstrate the practical potential of ideas borrowed from the field of metamaterials, we apply the concept of scattering-cancellation cloaking to design modified near-field probes that are effectively invisible and may enable non-perturbative near-field measurements. We then tackle a major challenge of linear passive cloaking devices, namely, their narrow bandwidth, and show that the “Bode-Fano limit” can be overcome through the design of active scattering-cancellation cloaks. We show that active cloaks indeed exhibit wider cloaking bandwidths; however, stability issues ultimately limit their performance. We then study another relevant application of the Bode-Fano bound in the context of broadband impedance matching of lossy films and solar cells, and we calculate the maximum solar power absorption for ultrathin solar cells made of different common materials and with arbitrary anti-reflection coatings. The second part of this dissertation focuses on wavefront manipulation for two relevant applications. We first propose a general platform, based on metasurfaces in waveguide networks, for analog optical computing. Based on this idea, we design compact devices for fractional calculus, optical PID controllers, and equation solvers. Finally, we design dielectric nonlocal metasurfaces that implement the transfer function of free space over a much shorter length. Such space-compression metasurfaces provide a solution to realize compact, fully solid-state, planar structures for focusing, imaging, and magnification. We believe the results of this dissertation may open many new opportunities for different engineering applications in electromagnetics, optics, and photonics.