Periodic structures have been utilized in many novel active devices due to their unique properties such as the presence of electromagnetic band-edges and band-gaps. Dispersion diagrams are associated to such structures with a unit cell repeated periodically and show the relation between frequency and periodic state phase shift or the eigen-states of the system. The dispersion characteristics of periodic structures can be engineered to exhibit exceptional modal characteristics, which can be exploited to design novel devices with improved features and enhanced performance. Distinct sorts of degeneracy may exist in periodic schemes where the eigen-states of the system coalesce and form a single degenerate periodic eigen-state. These phenomena can be classified as exceptional points of degeneracy (EPD) and is shown to have some fascinating features which make them desirable for a wide variety of applications including oscillators, amplifiers, lasers, and pulse compressors which are widely used in IoT scenarios. A special and well-known category of degeneracies in periodic structures is recognized as the fourth order degenerate band edge (DBE) where four periodic eigenstates coincide at the edge of the Brillouin zone. In this work we put the focus on two other significant types of EPDs which are the stationary inflection point (SIP, with third order degeneracy) and the sixth order degenerate band edge (6DBE). In particular, the 6DBE characteristics are expected to be noteworthy in enhancement of the Q-factor and in lowering the oscillation threshold in finite length resonators with gain compared to conventional uniform or homogeneous structures. Furthermore, a novel amplification regime based on special dispersion characteristics of an SIP, also called “frozen mode regime”, is proposed leading to a higher gain and larger gain-bandwidth product compared to conventional Pierce-type traveling wave tubes. Moreover, a novel design of a periodically coupled three-way microstrip waveguide is presented utilizing PT-glide symmetry by proper balancing of the loss/gain featuring third order EPDs. This latter concept could be advantageous in diverse applications including but not limited to distributed amplifiers and radiating arrays of antennas. The first physical realization and experimental demonstration of an exceptional point of sixth order (6DBE) in a triple ladder (three-way) microwave waveguide is provided; the unique properties of the 6DBE may be exploited in designing innovative high-Q resonators, oscillators, filters, and pulse shaping devices. An in-depth investigation of the modes and dispersion relations based on the state vectors and transfer matrix formalism is offered where we have also provided the analytical framework for the Puiseux fractional power series expansion of the system’s eigenvalues around the degeneracy condition that will be used to approximate the dispersion relation, density of states (DoS), and other important parameters. The analytical framework and physical concepts established in this thesis may be applied to a variety of structure designs and applications featuring degeneracy conditions of different orders and can be a useful tool in designing and evaluating novel passive and active devices.