  We review recent progress on cosmological issues and theoretical properties
of modified gravity theories. In particular, we explicitly explore the
conformal transformation, the Starobinsky inflation, and a unified scenario of
inflation and late time acceleration in $F(R)$ gravity and $F(T)$ gravity
(extended teleparallel gravity). Furthermore, we examine neutron stars and the
hyperon problem in $F(R)$ gravity. Moreover, for loop quantum cosmology (LQC),
the natures of finite-time future singularities in $F(T)$ gravity are
presented. In addition, we investigate $F(T)$ gravity theories from the
Kaluza-Klein (KK) and Randall-Sundrum (RS) theories.
