  In this paper, we investigate critical maps of the horizontal energy
functional $E_{H,\widetilde{H}}(f)$ for maps between two pseudo-Hermitian
manifolds $(M^{2m+1},H(M),J,\theta )$ and $(N^{2n+1},\widetilde{H}(N),
\widetilde{J},\widetilde{\theta})$. These critical maps are referred to as
$(H,\widetilde{H})$-harmonic maps. We derive a CR Bochner formula for the
horizontal energy density $|df_{H, \widetilde{H}}|^{2}$, and introduce a
Paneitz type operator acting on maps to refine the Bochner formula. As a
result, we are able to establish some Bochner type theorems for
$(H,\widetilde{H})$-harmonic maps. We also introduce
$(H,\widetilde{H})$-pluriharmonic, $(H,\widetilde{H})$-holomorphic maps between
these manifolds, which provide us examples of $(H,\widetilde{H})$-harmonic
maps. Moreover, a Lichnerowicz type result is established to show that foliated
$(H,\widetilde{ H})$-holomorphic maps are actually minimizers of
$E_{H,\widetilde{H}}(f)$ in their foliated homotopy classes. We also prove some
unique continuation results for characterizing either horizontally constant
maps or foliated $(H,\widetilde{H})$-holomorphic maps. Furthermore,
Eells-Sampson type existence results for $(H,\widetilde{H})$-harmonic maps are
established if both manifolds are compact Sasakian and the target is regular
with non-positive horizontal sectional curvature. Finally, we give a foliated
rigidity result for $(H,\widetilde{H})$-harmonic maps and Siu type strong
rigidity results for compact regular Sasakian manifolds with either strongly
negative horizontal curvature or adequately negative horizontal curvature.
