The past 30 years have seen a considerable effort on the part of statisticians to improve the design and accuracy of early-phase oncology trials. Some of this effort has been rewarded via successful implementation in actual trials, yet it would be fair to say that among clinicians, there remains some reluctance to fully embrace more efficient model-based approaches. One reason for such reticence is the difficulty in understanding exactly what is being offered by more modern designs. Although it is generally accepted that these designs offer improvements over the old standard 3 + 3 design, a new question has then to be addressed: How should we decide among the new proposals which one is the best for our purpose? In this study, we recall 15 designs that are currently proposed and in use. We show that among these 15 designs, many are operationally identical. These 15 designs reduce to three broad classes of designs. This review helps summarize their properties and differences and highlights that certain designs require ad hoc modifications to ensure satisfactory performance.