Since the proposal of the concept of topological superconductivity in condensed-matter physics, a new chapter has opened in the history of the development of topological superconductivity and Majorana zero modes (MZMs). Topological superconductors have become the main candidate materials for realizing topological quantum computing due to the Majorana zero-energy mode satisfying non-Abelian statistics on the boundary. In recent years, great progress has been achieved in both experiment and theory. The iron-based superconductors provide a single material with high-Tc superconductivity and Dirac topological surface states, which enable the easy detection and control of the MZMs in a single material at high temperature. Here, we review the current status of the exploration of MZMs in high-temperature iron-based thin film and bulk superconductors, especially the basic concepts and recent experimental results, paying particular attention to the observation of Dirac cone-type topological surface bands (nontrivial topology in the band structures) and the zero-bias conductance peak by STM/S.