Epithelial–mesenchymal transition (EMT) is a process that takes place during embryonic development, wound healing, and under some pathological processes, including fibrosis and tumor progression. The molecular changes occurring within epithelial cells during transformation to a mesenchymal phenotype have been well studied. However, to date, the mechanism of EMT induction remains to be fully elucidated. Recent findings in the field of intercellular communication have shed new light on this process and indicate the need for further studies into this important mechanism. New evidence supports the hypothesis that intercellular communication between mesenchymal stroma/stem cells (MSCs) and resident epithelial cells plays an important role in EMT induction. Besides direct interactions between cells, indirect paracrine interactions by soluble factors and extracellular vesicles also occur. Extracellular vesicles (EVs) are important mediators of intercellular communication, through the transfer of biologically active molecules, genetic material (mRNA, microRNA, siRNA, DNA), and EMT inducers to the target cells, which are capable of reprogramming recipient cells. In this review, we discuss the role of intercellular communication by EVs to induce EMT and the acquisition of stemness properties by normal and tumor epithelial cells.