  This paper gives a survey of known results concerning the Laplace transform
$$ L_k(s) := \int_0^\infty |\zeta(1/2+ ix)|^{2k}{\rm e}^{-sx}{\rm d} x \qquad(k
\in N, \R s > 0), $$ and the (modified) Mellin transform $$ {\cal Z}_k(s) :=
\int_1^\infty|\zeta(1/2+ ix)|^{2k}x^{-s}{\rm d} x\qquad(k\in N), $$ where the
integral is absolutely convergent for $\R s \ge c(k) > 1$. Also some new
results on these integral transforms of $|\zeta(1/2+ ix)|^{2k}$ are given,
which have important connections with power moments of the Riemann
zeta-function $\zeta(s)$.
