  Motivated by numerous questions in random geometry, given a smooth manifold
$M$, we approach a systematic study of the differential topology of Gaussian
random fields (GRF) $X:M\to \mathbb{R}^k$, that we interpret as random
variables with values in $\mathcal{C}^r(M, \mathbb{R}^k)$, inducing on it a
Gaussian measure.
  When the latter is given the weak Whitney topology, the convergence in law of
$X$ allows to compute the limit probability of certain events in terms of the
probability distribution of the limit. This is true, in particular, for the
events of a geometric or topological nature, like: "$X$ is transverse to $W$"
or "$X^{-1}(0)$ is homeomorphic to $Z$".
  We relate the convergence in law of a sequence of GRFs with that of their
covariance structures, proving that in the smooth case ($r=\infty$), the two
conditions coincide, in analogy with what happens for finite dimensional
Gaussian measures. We also show that this is false in the case of finite
regularity ($r\in\mathbb{N}$), although the convergence of the covariance
structures in the $\mathcal{C}^{r+2}$ sense is a sufficient condition for the
convergence in law of the corresponding GRFs in the $\mathcal{C}^r$ sense.
  We complement this study by proving an important technical tools: an infinite
dimensional, probabilistic version of the Thom transversality theorem, which
ensures that, under some conditions on the support, the jet of a GRF is almost
surely transverse to a given submanifold of the jet space.
