  In this talk, we'll present some recent results related to group actions in
several complex variables. We'll not aim at giving a complete survey about the
topic but giving some our own results and related ones. We'll divide the
results into two cases: compact and noncompact transformation groups. We
emphasize some essential differences between the two cases. In the compact
case, we'll mention some results about schlichtness of envelopes of holomorphy
and compactness of automorphism groups of some invariant domains. In the
noncompact case, we'll present our solution of the longstanding problem -- the
so-called extended future tube conjecture which asserts that the extended
future tube is a domain of holomorphy. Invariant version of Cartan's lemma
about extension of holomorphic functions from the subvarities in the sense of
group actions will be also mentioned.
