  In this paper the focus is on inflationary dynamics in the context of
Einstein Gauss-Bonnet gravitational theories. We investigate the implications
of the slow-roll condition on the slow-roll indices and we investigate how the
inflationary dynamical evolution is affected by the presence of the
Gauss-Bonnet coupling to the scalar field. For exemplification of our analysis
we investigate how the dynamics of inflationary cubic, quartic order and also
exponential scalar potentials are affected by the non-trivial Gauss-Bonnet
coupling to the scalar field. As we demonstrate it is possible to obtain a
viable phenomenology compatible with the observational data, although the
canonical scalar field theory with cubic and quartic order potentials does not
yield phenomenologically acceptable results. In addition, with regard to the
exponential potential example, the Einstein Gauss-Bonnet extension of the
single canonical scalar field model has an inherent mechanism that can trigger
the graceful exit from inflation. Furthermore we introduce a bottom-up
reconstruction technique, in the context of which by fixing the
tensor-to-scalar ratio and the Hubble rate as a function of the $e$-foldings
number, one is capable of reproducing the Einstein Gauss-Bonnet theory which
generates the aforementioned quantities. We illustrate how the method works by
using some relatively simple examples.
