Scattering of electromagnetic waves lies at the heart of most experimental techniques over nearly the entire electromagnetic spectrum, ranging from radio waves to optics and X-rays. Hence, deep insight into the basics of scattering theory and understanding the peculiar features of electromagnetic scattering is necessary for the correct interpretation of experimental data and an understanding of the underlying physics. Recently, a broad spectrum of exceptional scattering phenomena attainable in suitably engineered structures has been predicted and demonstrated. Examples include bound states in the continuum, exceptional points in PT-symmetrical non-Hermitian systems, coherent perfect absorption, virtual perfect absorption, nontrivial lasing, non-radiating sources, and others. In this paper, we establish a unified description of such exotic scattering phenomena and show that the origin of all these effects can be traced back to the properties of poles and zeros of the underlying scattering matrix. We provide insights on how managing these special points in the complex frequency plane provides a powerful approach to tailor unusual scattering regimes.