For elements of an infinite planar phased array in free space and an infinite conductor-backed free standing planar phased array, fundamental bandwidth bounds are derived from the optical theorem for periodic scatterers. The bounds are based on Gustafsson’s theory of fundamental bandwidth limits for electrically small antennas of arbitrary shape, but extended to periodic configurations. The element bandwidth bound is found to be a function of the strengths of the induced static dipole moments in constant background fields as well as the unit-cell dimensions and the scan angle. Explicit bandwidth bound expressions are found for narrowband and ultrawideband array cases. For elements of an infinite conductor-backed planar phased array with a dielectric substrate, fundamental bandwidth bounds are derived by Doane based on the Fano’s method. A general conductor-backed array is modeled as transmission line terminate with a shorted load. The input reflection coefficient is expanded in long wavelength limit and Fano’s method is used to determine a limit for the frequency integral of the reflection coefficient. This yields the bandwidth bound for the array. In this dissertation, the reflection coefficients were expanded into high order terms. Then, Fano limit is invoked to have the higher-order bandwidth bound. Such bandwidth bound depends on the physical features (scan angle, ground separation, unit-cell area) and electric feature (dipole strength). All bandwidth bound are numerically tested using example array designs. The following time-harmonic analysis, an e^jωt time convention is assumed and suppressed. Peak phasors are used for source and field quantities.