  We present a comprehensive analysis of a linear growth model, which combines
the characteristic features of the Edwards--Wilkinson and noisy Mullins
equations. This model can be derived from microscopics and it describes the
relaxation and growth of surfaces under conditions where the nonlinearities can
be neglected. We calculate in detail the surface width and various correlation
functions characterizing the model. In particular, we study the crossover
scaling of these functions between the two limits described by the combined
equation. Also, we study the effect of colored and conserved noise on the
growth exponents, and the effect of different initial conditions. The
contribution of a rough substrate to the surface width is shown to decay
universally as $w_i(0) (\xi_s/\xi(t))^{d/2}$, where $\xi(t) \sim t^{1/z}$ is
the time--dependent correlation length associated with the growth process,
$w_i(0)$ is the initial roughness and $\xi_s$ the correlation length of the
substrate roughness, and $d$ is the surface dimensionality. As a second
application, we compute the large distance asymptotics of the height
correlation function and show that it differs qualitatively from the functional
forms commonly used in the intepretation of scattering experiments.
