  In this paper, we give a brief survey of recent developments on Noether's
problem and rationality problem for multiplicative invariant fields including
author's recent papers Hoshi [Hos15] about Noether's problem over Q, Hoshi,
Kang and Kunyavskii [HKK13], Chu, Hoshi, Hu and Kang [CHHK15], Hoshi [Hos16]
and Hoshi, Kang and Yamasaki [HKY16] about Noether's problem over C, and Hoshi,
Kang and Kitayama [HKK14] and Hoshi, Kang and Yamasaki [HKY] about rationality
problem for multiplicative invariant fields.
