A review is made of the current status of investigations of the Josephson effect in hightemperature superconducting materials and in structures based on them. The results are given of experimental investigations of the influence of various factors on superconducting properties of surface layers of high-temperature materials, such as the conditions during synthesis and cleaning of their surface, and also data on the ohmic contacts. This information is used to analyze the experimental results obtained for Josephson junctions between high- and low-temperature superconductors and between two high-temperature superconductors. The following junctions are considered: point contacts, tunnel junctions, SNS structures with direct conduction, junctions at internal grain boundaries in high-temperature superconducting ceramics and films, and also junctions at microcracks in high-temperature superconducting crystals. These results are compared with the predictions of theoretical models of Josephson junctions developed on the basis of the BCS theory. It is concluded that the available data are insufficient not only to determine the validity of the BCS theory in the description of high-temperature superconductors, but also to provide a reliable qualitative identification of the physical structure of the junctions. Possible topics for further experimental investigations of the Josephson effect in high-temperature superconducting junctions with direct and tunnel conduction are suggested.