Chronic myeloid leukemia (CML) is caused by the acquisition of the tyrosine kinase BCR-ABL1 in a hemopoietic stem cell, transforming it into a leukemic stem cell (LSC) that self-renews, proliferates, and differentiates to give rise to a myeloproliferative disease. Although tyrosine kinase inhibitors (TKIs) that target the kinase activity of BCR-ABL1 have transformed CML from a once-fatal disease to a manageable one for the vast majority of patients, only ∼10% of those who present in chronic phase (CP) can discontinue TKI treatment and maintain a therapy-free remission. Strong evidence now shows that CML LSCs are resistant to the effects of TKIs and persist in all patients on long-term therapy, where they may promote acquired TKI resistance, drive relapse or disease progression, and inevitably represent a bottleneck to cure. Since their discovery in patients almost 2 decades ago, CML LSCs have become a well-recognized exemplar of the cancer stem cell and have been characterized extensively, with the aim of developing new curative therapeutic approaches based on LSC eradication. This review summarizes our current understanding of many of the pathways and mechanisms that promote the survival of the CP CML LSCs and how they can be a source of new gene coding mutations that impact in the clinic. We also review recent preclinical approaches that show promise to eradicate the LSC, and future challenges on the path to cure.