In this chapter, I delve into complex light by treating the most important beams of light that are useful for research in many areas of science and technology. It includes a description of the functional forms of the modes and their parameters, which lead to better resolution and deeper penetration. I will first cover the fundamentals of Gaussian beams, which are naturally generated by a laser, and which constitute the workhorse of laser-based research on multiphoton and confocal microscopes, and in manipulation of matter via optical forces. I will then continue with interesting optical realizations that carry a new range of physical phenomena, such as optical vortices and orbital angular momentum in the case of Laguerre-Gauss beams; or diffraction-less propagation and self-reconstructing beams in the case with Bessel and Airy beams; or space-variant polarization and singularities in polarization in the case of vector and Poincaré beams. The different sections also discuss and cite the various methods for producing these beams. It is hoped that they will stimulate new research in imaging, manipulation, and new areas where complex light has yet to make an impact, leading to the discovery of new phenomena and applications. 2.2 Gaussian BeamsIn this section, we discuss optical beams that are described by a Gaussian intensity profile. This is the case of laser beams. Thus it is important to begin with a discussion of the fundamental Gaussian beam and its properties. We then follow with sections describing important high-order Gaussian beams, which are also beams that can be derived from the fundamental one. 2.2.1 Fundamental Gaussian BeamsWe begin by deriving the equation of the fundamental Gaussian beam. This treatment is also covered in standard textbooks on lasers [40, 45]. In classical electromagnetism, light is represented by an electromagnetic wave, with electric and magnetic fields that obey Maxwell’s equations. Since the electric and magnetic fields are related to each other, it is usual to express light in terms of one of them: the electric field.