Cancers remain the second leading cause of mortality in the world. Preclinical and clinical studies point an important role of cancer/leukaemia stem cells (CSCs/LSCs) in the colonisation at secondary organ sites upon metastatic spreading, although the precise mechanisms for specific actions are still not fully understood. Reviewing the present knowledge on the crucial role of CSCs/LSCs, their plasticity, and population heterogeneity in treatment failures in cancer patients is timely. Standard chemotherapy, which acts mainly on rapidly dividing cells, is unable to adequately affect CSCs with a low proliferation rate. One of the proposed mechanisms of CSC resistance to anticancer agents is the fact that these cells can easily shift between different phases of the cell cycle in response to typical cell stimuli induced by anticancer drugs. In this work, we reviewed the recent studies on CSC/LSC alterations associated with disease recurrence, and we systematised the functional assays, markers, and novel methods for CSCs screening. This review emphasises CSCs’ involvement in cancer progression and metastasis, as well as CSC/LSC targeting by synthetic and natural compounds aiming at their elimination or modulation of stemness properties.