  In this paper we discuss some basic properties of octonionic Bergman and
Hardy spaces. In the first part we review some fundamental concepts of the
general theory of octonionic Hardy and Bergman spaces together with related
reproducing kernel functions in the monogenic setting. We explain how some of
the fundamental problems in well-defining a reproducing kernel can be overcome
in the non-associative setting by looking at the real part of an appropriately
defined para-linear octonion-valued inner product. The presence of a weight
factor of norm $1$ in the definition of the inner product is an intrinsic new
ingredient in the octonionic setting. Then we look at the slice monogenic
octonionic setting using the classical complex book structure. We present
explicit formulas for the slice monogenic reproducing kernels for the unit
ball, the right octonionic half-space and strip domains bounded in the real
direction. In the setting of the unit ball we present an explicit sequential
characterization which can be obtained by applying the special Taylor series
representation of the slice monogenic setting together with particular
octonionic calculation rules that reflect the property of octonionic
para-linearity.
