We formulated a global Gan-Gross-Prasad conjecture for general spin groups. That is, we formulated a conjecture on a relation between periods of certain automorphic forms on $GSpin_{n+1} \times GSpin_n$ along the diagonal subgroup $GSpin_n$ and some L-values. To support the conjecture, we will discuss work that shows the conjecture holds for $n=2$ and $3$ and for certain cases for $n=4$.