  In this paper we investigate how to realize various quite well known
cosmological bouncing models in the context of the recently developed
unimodular $F(R)$ gravity. Particularly, we shall study the matter bounce
scenario, the singular bounce, the superbounce and a symmetric bounce scenario.
We present the behavior of the Hubble radius for each of the bouncing models we
shall take into account and we investigate which era of the bouncing model is
responsible for the cosmological perturbations. As we shall demonstrate, the
various bouncing models do not behave in the same way, so the cosmological
perturbations for each model may correspond to a different era, in comparison
to other models. Also we present which unimodular $F(R)$ gravity realizes each
model. We also show that Newton's law is not modified in the unimodular $F(R)$
gravity, which also is proven to be a ghost-free theory, and in addition we
discuss the matter stability issue. Finally, we demonstrate how it is possible
to solve a cosmological constant problem in the context of unimodular $F(R)$
gravity.
