  We study the $2k$-th moment at the central point of the family of symmetric
square $L$-functions attached to holomorphic Hecke cusp forms of level one,
weight $\kappa$. We establish sharp lower bounds for all real $k \geq 1/2$
unconditionally. Assuming the truth of the generalized Riemann hypothesis, we
also obtain sharp lower bounds for all real $0 \leq k < 1/2$ and sharp upper
bounds for all real $k \geq 0$.
