Natural rubber (NR) remains an indispensable raw material with unique properties that is used in the manufacture of a large number of products and the global demand for it is growing every year. The only industrially important source of NR is the tropical tree Hevea brasiliensis (Willd. ex A.Juss.) Müll.Arg., thus alternative sources of rubber are required. For the temperate zone, the most suitable source of high quality rubber is the Russian (Kazakh) dandelion Taraxacum kok-saghyz L.E. Rodin (TKS). An obstacle to the widespread industrial cultivation of TKS is its high heterozygosity, poor growth energy, and low competitiveness in the field, as well as inbreeding depression. Rapid cultivation of TKS requires the use of modern technologies of marker-assisted and genomic selection, as well as approaches of genetic engineering and genome editing. This review is devoted to describing the progress in the field of molecular genetics, genomics, and genetic engineering of TKS. Sequencing and annotation of the entire TKS genome made it possible to identify a large number of SNPs, which were subsequently used in genotyping. To date, a total of 90 functional genes have been identified that control the rubber synthesis pathway in TKS. The most important of these proteins are part of the rubber transferase complex and are encoded by eight genes for cis-prenyltransferases (TkCPT), two genes for cis-prenyltransferase-like proteins (TkCPTL), one gene for rubber elongation factor (TkREF), and nine genes for small rubber particle proteins (TkSRPP). In TKS, genes for enzymes of inulin metabolism have also been identified and genome-wide studies of other gene families are also underway. Comparative transcriptomic and proteomic studies of TKS lines with different accumulations of NR are also being carried out, which help to identify genes and proteins involved in the synthesis, regulation, and accumulation of this natural polymer. A number of authors already use the knowledge gained in the genetic engineering of TKS and the main goal of these works is the rapid transformation of the TKS into an economically viable rubber crop. There are no great successes in this area so far, therefore work on genetic transformation and genome editing of TKS should be continued, considering the recent results of genome-wide studies.