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Highlights • Constructs the Hertzian piezoelectric contact model by cumulative superposition for a rigid punch of general profile. • Establishes the JKR-type adhesive contact model for a rigid conducting punch of power-law profile with use of the developed equivalent energy release rate approach. • Develops a series of closed-form analytical solutions which include the classical Hertz- n and JKR- n solutions as special cases. • Reveals the effect of shape index on indentation responses of piezoelectric materials in the non-adhesion and JKR adhesion regimes. Abstract The present paper provides a theoretical study on the axisymmetric contact and adhesive behaviors of a transversely isotropic piezoelectric half-space under a rigid punch, with an emphasis on Hertzian contact and JKR-type adhesion solutions for the conducting punch of power-law profile. The Hertzian piezoelectric contact model and JKR-type adhesion model are established by cumulative superposition and equivalent energy release rate approaches, respectively. Closed-form expressions are derived for the applied load, indentation depth, distributions of contact traction and surface displacement, which are able to link existing solutions for non-adhesive and adhesive contact problems for transversely isotropic and isotropic elastic materials, and to include the corresponding Hertz- n and JKR- n solutions as special cases. Similar mathematical forms are found between the elastic solids and the piezoelectric materials under a conducting grounded punch except for different material moduli. These results can serve as benchmarks for checking the validity of numerical methods. [ABSTRACT FROM AUTHOR]