  In this paper, we consider a system of $k$ second order non-linear stochastic
partial differential equations with spatial dimension $d \geq 1$, driven by a
$q$-dimensional Gaussian noise, which is white in time and with some spatially
homogeneous covariance. The case of a single equation and a one-dimensional
noise, has largely been studied in the literature. The first aim of this paper
is to give a survey of some of the existing results. We will start with the
existence, uniqueness and H\"older's continuity of the solution. For this, the
extension of Walsh's stochastic integral to cover some measure-valued
integrands will be recalled. We will then recall the results concerning the
existence and smoothness of the density, as well as its strict positivity,
which are obtained using techniques of Malliavin calculus. The second aim of
this paper is to show how these results extend to our system of SPDEs. In
particular, we give sufficient conditions in order to have existence and
smoothness of the density on the set where the columns of the diffusion matrix
span $\R^k$. We then prove that the density is strictly positive in a point if
the connected component of the set where the columns of the diffusion matrix
span $\R^k$ which contains this point has a non void intersection with the
support of the law of the solution. We will finally check how all these results
apply to the case of the stochastic heat equation in any space dimension and
the stochastic wave equation in dimension $d\in \{1,2,3\}$.
