We introduce a conformally invariant nonlinear sigma model on the bulk of contact manifolds with boundary condition on the Legendrian links in any odd dimension. We call any finite energy solution a contact instanton. We also explain its Hamiltonian-perturbed equation and establish the Gromov-Floer-Hofer type convergence result for (Hamiltonian-perturbed) contact instantons of finite energy and construct its compactification of the moduli space. Then we explain how we can apply this analytical machinery in the study of contact topology and contact Hamiltonian dynamics. As an example, we explain our proof of a conjecture of Sandon and Shelukhin on the translated points of contactomorphisms on compact contact manifold. This is the contact analog to the Arnold's conjecture type in symplectic geometry.