In recent years, genome editing methods have become an integral part of the genetic engineering toolset that allows for making targeted changes to plant genomes, both in the case of single-gene mutations and multiplex modifications. These technologies were mostly proven effective for editing nuclear genomes. However, plastids, the best-known example of which is chloroplasts, have their own genome (plastome), which is also available for various genetic manipulations, including editing. Despite the fact that the modification of plastomes represents a very promising task for modern biotechnology, the structure of plastids and the peculiarities of their genome organization require the specific adaptation of genome editing methods. This applies to both the design of genetic constructs and methods of their delivery to plastids. The article provides an overview of the current state of research in the field of plastid genome editing with chloroplasts taken as an example. We consider the possibilities of using programmable genome-editing technologies, analyze their effectiveness, limitations, and problems caused by the structural features of these organelles, and their genome organization. We discuss the results of the first successful experiments in this field and try to assess the prospects for the development of tools and methods for increasing the efficiency and the specificity of this biotechnological platform.