This is the first account of the history of our understanding of, and ability to model, solidification microstructures. Its objective is to retrace the scientific steps made, from the earliest observations of the eighteenth century to our present-day understanding of dendrites and eutectics. Because of the abundance of information, especially that added during the present century, sub-division was essential: this being the first of three articles. They cover dendrites and cells from 1700 to 2000, and then from 2001 to 2015 and finally eutectics and peritectics from 1700 to 2015. The authors have striven always to identify the genesis of every advance made, being aware that such a compact history must leave many worthy contributions by the wayside; others will doubtless complete the history. This review shows how cross-fertilisation between theory and experiment, and basic and applied research led to both the posing and answering of challenging fundamental questions, thus rewarding society with beneficial results.