  Photoproduction of $J/\psi$ is considered including the color-octet
contributions from the various partial wave states, $^{2S+1}L_J$ = $^1S_0$,
$^3S_1$ and $^3P_J$. The production cross section depends on three new
nonperturbative parameters defined in NRQCD, called the color-octet matrix
elements. Using the color-octet matrix elements determined by fitting the
$J/\psi$ production at the Tevatron, we find that the color-octet($c\bar{c})
_{8}({^1S_0}~{\rm and~}~{^3P_J})$ contributions to the $J/\psi$
photoproductions at the fixed target experiments and HERA are too large
compared to the data on $\sigma (\gamma + p \to J/\psi +X)$ in the forward
direction, the $z$ distribution of $J/\psi$. The $P_T^2$ distribution of
$J/\psi$ and the total inelastic $J/\psi$ production rate as a function of
$\sqrt{s_ {\gamma p}}$ are predicted including color-octet contributions. We
also briefly digress on the $B \to J/\psi + X$ and observe the similar
situation. This may be an indication that the color-octet matrix elements
determined from the $J/\psi$ production at the Tevatron, especially $<0 | {\cal
O}_{8}^{\psi} ({^1S_0}) | 0>$ and $<0 | {\cal O}_{8}^{\psi} ({^3P_J}) | 0>$,
might have been overestimated by an order of magnitude.
