In 1980 Charles Leedham-Green and Mike Newman published the paper: "Space groups and groups of prime-power order I". The main goal of the paper was to introduce five conjectures, namely Conjectures A-E, in descending strength. The rest of the paper was fairly technical results (as far as I understand). These five conjectures are the coclass conjectures.
The conjectures attracted a lot of attention and many people were involved in proving them. I do not recall the whole history, but Conjecture E was proved by Leedham-Green, McKay and Plesken. Then results of Donkin and Leedham-Green proved Conjecture C for $p>3$ and later on Conjecture A also for $p>3$. Afterwards, Shalev and Zelmanov proved all the conjectures for all $p$'s. Finally Shalev gave an effective proof of the conjectures.
These days there is still research done on the structure of groups of finite coclass.