  We introduce a broad class of self-similar processes $\{Z(t),t\ge 0\}$ called
generalized Hermite process. They have stationary increments, are defined on a
Wiener chaos with Hurst index $H\in (1/2,1)$, and include Hermite processes as
a special case. They are defined through a homogeneous kernel $g$, called
"generalized Hermite kernel", which replaces the product of power functions in
the definition of Hermite processes. The generalized Hermite kernels $g$ can
also be used to generate long-range dependent stationary sequences forming a
discrete chaos process $\{X(n)\}$. In addition, we consider a
fractionally-filtered version $Z^\beta(t)$ of $Z(t)$, which allows $H\in
(0,1/2)$. Corresponding non-central limit theorems are established. We also
give a multivariate limit theorem which mixes central and non-central limit
theorems.
