  Faltings initiated in 2005 a p-adic analogue of the (complex) Simpson
correspondence whose construction has been taken up by various authors,
according to several approaches. Following the one we initiated previously, we
present an overview of a new monograph developing new features of the p-adic
Simpson correspondence, inspired by our construction of the relative Hodge-Tate
spectral sequence. First, we address the connection to Hodge-Tate local
systems. Second, we establish the functoriality of the p-adic Simpson
correspondence by proper direct image. Along the way, we expand the scope of
our original construction.
