  Let $S=XX^T$ be the (unscaled) sample covariance matrix where $X$ is a real
$p \times n$ matrix with independent entries. It is well known that if the
entries of $X$ are independent and identically distributed (i.i.d.) with enough
moments and $p/n \to y\neq 0$, then the limiting spectral distribution (LSD) of
$\frac{1}{n}S$ converges to a Mar$\check{\text{c}}$enko-Pastur law. Several
extensions of this result are also known. We prove a general result on the
existence of the LSD of $S$ in probability or almost surely, and in particular,
many of the above results follow as special cases. At the same time several new
LSD results also follow from our general result.
  The moments of the LSD are quite involved but can be described via a set of
partitions. Unlike in the i.i.d. entries case, these partitions are not
necessarily non-crossing, but are related to the special symmetric partitions
which are known to appear in the LSD of (generalised) Wigner matrices with
independent entries.
  We also investigate the existence of the LSD of $S_{A}=AA^T$ when $A$ is the
$p\times n$ symmetric or the asymmetric version of any of the following four
random matrices: reverse circulant, circulant, Toeplitz and Hankel. The LSD of
$\frac{1}{n}S_{A}$ for the above four cases have been studied by Bose,
Gangopadhyay and Sen in 2010, when the entries are i.i.d. We show that under
some general assumptions on the entries of $A$, the LSD of $S_{A}$ exists and
this result generalises the existing results significantly.
