Is there a classification of finite nonabelian 2groups of exponent 4?
What about, finite nonabelian 3groups of exponent 3?
Is there a classification of finite nonabelian 2groups of exponent 4? What about, finite nonabelian 3groups of exponent 3? 


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 exponent4 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 $3^{m + \binom{m}{2} + \binom{m}{3}}$. 


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

