For millennia, people have cut and joined different plants together through a process known as grafting. The severed tissues adhere, the cells divide and the vasculature differentiates through a remarkable process of regeneration between two genetically distinct organisms as they become one. Grafting is becoming increasingly important in horticulture where it provides an efficient means for asexual propagation. Grafting also combines desirable roots and shoots to generate chimeras that are more vigorous, more pathogen resistant and more abiotic stress resistant. Thus, it presents an elegant and efficient way to improve plant productivity in vegetables and trees using traditional techniques. Despite this horticultural importance, we are only beginning to understand how plants regenerate tissues at the graft junction. By understanding grafting better, we can shed light on fundamental regeneration pathways and the basis for self/non-self recognition. We can also better understand why many plants efficiently graft whereas others cannot, with the goal of improving grafting so as to broaden the range of grafted plants to create even more desirable chimeras. Here, I review the latest findings describing how plants graft and provide insight into future directions in this emerging field.