  We used the mark weighted correlation functions (MCFs), $W(s)$, to study the
large scale structure of the Universe. We studied five types of MCFs with the
weighting scheme $\rho^\alpha$, where $\rho$ is the local density, and $\alpha$
is taken as $-1,\ -0.5,\ 0,\ 0.5$, and 1. We found that different MCFs have
very different amplitudes and scale-dependence. Some of the MCFs exhibit
distinctive peaks and valleys that do not exist in the standard correlation
functions. Their locations are robust against the redshifts and the background
geometry, however it is unlikely that they can be used as ``standard rulers''
to probe the cosmic expansion history. Nonetheless we find that these features
may be used to probe parameters related with the structure formation history,
such as the values of $\sigma_8$ and the galaxy bias. Finally, after conducting
a comprehensive analysis using the full shapes of the $W(s)$s and $W_{\Delta
s}(\mu)$s, we found that, combining different types of MCFs can significantly
improve the cosmological parameter constraints. Compared with using only the
standard correlation function, the combinations of MCFs with $\alpha=0,\ 0.5,\
1$ and $\alpha=0,\ -1,\ -0.5,\ 0.5,\ 1$ can improve the constraints on
$\Omega_m$ and $w$ by $\approx30\%$ and $50\%$, respectively. We find highly
significant evidence that MCFs can improve cosmological parameter constraints.
