  We discuss two-photon and hadronic production of $B_c$ mesons in
nonrelativistic bound state approximation and to lowest order in the coupling
constants $\alpha$ and $\alpha_s$. It is shown that in photon-photon
collisions, heavy quark fragmentation is dominated by recombination of $\bar b$
and $c$ quarks up to the highest accessible transverse momenta. In contrast, in
hadroproduction, which at high energies mainly involves gluon--gluon
collisions, the fragmentation mechanism dominates at transverse momenta $p_T >
m_{B_c}$, providing a simple and satisfactory approximation of the complete
$O(\alpha_s^4)$ results in the high-$p_T$ regime. Contradictions in previous
publications on hadroproduction of $B_c$ mesons are clarified. We also present
predictions for cross sections and differential distributions at present and
future accelerators.
