  We evaluate the threshold matrix-element for the reaction $pp \to pp\pi^0$ in
a fully relativistic Feynman diagrammatic approach. We employ a simple
effective range approximation to take care of the S-wave $pp$ final-state
interaction. The experimental value for the threshold amplitude ${\cal A} =
(2.7 - i 0.3)$ fm$^4$ can be reproduced by contributions from tree level chiral
(long-range) pion exchange and short-range effects related to vector meson
exchanges, with $\omega$ exchange giving the largest individual contribution.
Pion loop effects appear to be small. We stress that the commonly used heavy
baryon formalism is not applicable in the NN-system above the pion production
threshold due to the large external momentum, $|\vec p | \simeq \sqrt
{Mm_\pi}$, with $M$ and $m_\pi$ the nucleon and the pion mass, respectively. We
furthermore investigate the reaction $pp\to p n \pi^+$ near threshold within
the same approach. We extract from the data the triplet threshold amplitude as
${\cal B}= (2.8 -i 1.5)$ fm$^4$. Its real part can be well understood from
(relativistic) tree level meson-exchange diagrams. In addition, we investigate
the process $pp \to pp \eta$ near threshold. We use a simple factorization
ansatz for the $pp\eta$ final-state interaction and extract from the data the
modulus of the threshold amplitude, $|{\cal C}| = 1.32 $fm$^4$. With $g_{\eta
N}=5.3$, this value can be reproduced by (relativistic) tree level
meson-exchange diagrams and $\eta$-rescattering, whose strength is fixed by the
$\eta N$ scattering length. We also comment on the recent near threshold data
for $\eta'$-production.
