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We propose a conductive carbon nanotube (CNT) sheet to realize conformal antennas on polymer substrates. Polymer-ceramic composites (rubber-like structures) have good RF (high dielectric constant and low loss tangent) and desirable mechanical properties (conformal, flexible and lightweight). However, there is a challenge in printing metallization circuits on polymer substrates due to their hydrophobic nature. Also, they are associated with low metal-polymer adhesion, causing peeling under stain or tensile stresses. To address these issues, in this paper, we consider the approach of embedding high density vertically-aligned carbon nanotubes within the polymer composite to achieve a CNT sheet having high structural compatibility. We present the fabrication process to achieve high conductivity CNT sheets and construct a sample polymer-CNT patch antenna, yielding a 5.6 dB gain. This is only 0.8 dB lower than that of an ideal patch made of perfect electric conductor (PEC). Strain and tensile tests are also carried out to evaluate electrical performance of the polymer-CNT sheet as it is bent and stretched. Our measurements show that the proposed conductive polymer-CNT sheet is highly flexible and preserves good conductivity under small bending and stretching. The CNT sheet retains acceptable performances even after 100° bending and 13% stretching. The proposed polymer-CNT sheets are well suited for load bearing antenna applications.