  We introduce state-space models where the functionals of the observational
and the evolutionary equations are unknown, and treated as random functions
evolving with time. Thus, our model is nonparametric and generalizes the
traditional parametric state-space models. This random function approach also
frees us from the restrictive assumption that the functional forms, although
time-dependent, are of fixed forms. The traditional approach of assuming known,
parametric functional forms is questionable, particularly in state-space
models, since the validation of the assumptions require data on both the
observed time series and the latent states; however, data on the latter are not
available in state-space models.
  We specify Gaussian processes as priors of the random functions and exploit
the "look-up table approach" of \ctn{Bhattacharya07} to efficiently handle the
dynamic structure of the model. We consider both univariate and multivariate
situations, using the Markov chain Monte Carlo (MCMC) approach for studying the
posterior distributions of interest. In the case of challenging multivariate
situations we demonstrate that the newly developed Transformation-based MCMC
(TMCMC) of \ctn{Dutta11} provides interesting and efficient alternatives to the
usual proposal distributions. We illustrate our methods with a challenging
multivariate simulated data set, where the true observational and the
evolutionary equations are highly non-linear, and treated as unknown. The
results we obtain are quite encouraging. Moreover, using our Gaussian process
approach we analysed a real data set, which has also been analysed by
\ctn{Shumway82} and \ctn{Carlin92} using the linearity assumption. Our analyses
show that towards the end of the time series, the linearity assumption of the
previous authors breaks down.
