  We propose a bigravity analogue of the $F(R)$ gravity. Our construction is
based on recent ghost-free massive bigravity where additional scalar fields are
added and the corresponding conformal transformation is implemented. It turns
out that $F(R)$ bigravity is easier to formulate in terms of the auxiliary
scalars as the explicit presentation in terms of $F(R)$ is quite cumbersome.
The consistent cosmological reconstruction scheme of $F(R)$ bigravity is
developed in detail, showing the possibility to realize nearly arbitrary
physical universe evolution with consistent solution for second metric. The
examples of accelerating universe which includes phantom, quintessence and
$\Lambda$CDM acceleration are worked out in detail and their physical
properties are briefly discussed.
