High Impedance Surfaces (HISs), which have been investigated extensively, have proven to be very efficient ground planes for low profile antenna applications due to their unique reflection phase characteristics. Another emerging research field among the microwave and antenna technologies is the design of flexible antennas and microwave circuits to be utilized in conformal applications. The combination of those two research topics gives birth to a third one, namely the design of Conformal or Flexible HISs (FHISs), which is the main subject of this dissertation. The problems associated with the FHISs are twofold: characterization and physical realization. The characterization involves the analysis of scattering properties of FHISs in the presence of plane wave and localized sources. For this purpose, an approximate analytical method is developed to characterize the reflection properties of a cylindrically curved FHIS. The effects of curvature on the reflection phase of the curved FHISs are examined. Furthermore, the effects of different types of currents, specifically the ones inherent to finite sized periodic structures, on the reflection phase characteristics are observed. After the reflection phase characterization of curved HISs, the performance of dipole antennas located in close proximity to a curved HIS are investigated, and the results are compared with the flat case. Different types of resonances that may occur for such a low-profile antenna application are discussed. The effects of curvature on the radiation performance of antennas are examined. Commercially available flexible materials are relatively thin which degrades the bandwidth of HISs. Another practical aspect, which is related to the substrate thickness, is the compactness of the surface. Because of the design limitations of conventional HISs, it is not possible to miniaturize the HIS and increase the bandwidth, simultaneously. To overcome this drawback, a novel HIS is proposed with a periodically perforated ground plane. Copper plated through holes are extremely vulnerable to bending and should be avoided at the bending parts of flexible circuits. Fortunately, if designed properly, the perforations on the ground plane may result in suppression of surface waves. Hence, metallic posts can be eliminated without hindering the surface wave suppression properties of HISs.