  We study optical phenomena related to appearance of Keplerian accretion discs
orbiting Kerr superspinars predicted by the string theory. The superspinar
exterior is described by the standard Kerr naked singularity geometry breaking
the black hole limit on the internal angular momentum (spin). We construct
local photon escape cones for a variety of orbiting sources that enable to
determine the superspinars silhouette in the case of distant observers. We show
that the superspinar silhouette depends strongly on the assumed edge where the
external Kerr spacetime is joined to the internal spacetime governed by the
string theory and significantly differs from the black hole silhouette. The
appearance of the accretion disc is strongly dependent on the value of the
superspinar spin in both their shape and frequency shift profile. Apparent
extension of the disc grows significantly with growing spin, while the
frequency shift grows with descending spin. This behavior differs substantially
from appearance of discs orbiting black holes enabling thus, at least in
principle, to distinguish clearly the Kerr superspinars and black holes. In
vicinity of a Kerr superspinar the non-escaped photons have to be separated to
those captured by the superspinar and those being trapped in its strong
gravitational field leading to self-illumination of the disc that could even
influence its structure and causes self-reflection effect of radiation of the
disc. The amount of trapped photons grows with descending of the superspinar
spin. We thus can expect significant self-illumination effects in the field of
Kerr superspinars with near-extreme spin $a \sim 1$.
