TITLE: Groups of exponent 4
QUESTION [7 upvotes]: Is there a classification of finite nonabelian 2-groups of exponent 4?
What about, finite nonabelian 3-groups of exponent 3?

REPLY [18 votes]: There is no classification of finite groups of exponent 4. You might find this paper interesting - it contains lots of information about how the group Burnside group $B(m,4)$ grows (all $m$-generator exponent-4 groups are quotients of this group).
There is also no classification of finite groups of exponent 3. However it is known that these groups must be $2$-Engel and class three. Furthermore in this case the precise size of the corresponding Burnside group is known: $B(m,3)$ is a finite group of size $3^{m + \binom{m}{2} + \binom{m}{3}}$.

REPLY [9 votes]: What about, finite nonabelian 3-groups
  of exponent 3?

Those are all quotients of the Burnside group $B(m,3)$ for some value for $m$.