  Let $\Omega$ be a class of ${\rm C^*}$-algebras. In this paper, we study a
class of not necessarily unital generalized tracial approximation ${\rm
C^*}$-algebras, and the class of simple ${\rm C^*}$-algebras which can be
generally tracially approximated by ${\rm C^*}$-algebras in $\Omega$, denoted
by ${\rm gTA}\Omega$. Let $\Omega$ be a class of unital ${\rm C^*}$-algebras
and let $A$ be a simple unital ${\rm C^*}$-algebra. Then $A\in {\rm
gTA}\Omega$, if, and only if, $A\in {\rm WTA}\Omega$ (where ${\rm TA}\Omega$ is
the class of weakly tracially approximable unital ${\rm C^*}$-algebras
introduced by Elliott, Fan, and Fang).Consider the class of ${\rm
C^*}$-algebras which are tracially $\mathcal{Z}$-absorbing (or are of tracial
nuclear dimension at most $n$, or are $m$-almost divisible, or have the
property $\rm SP$). Then $A$ is tracially $\mathcal{Z}$-absorbing
(respectively, has tracial nuclear dimension at most $n$, is weakly ($n,
m$)-almost divisible, has the property $\rm SP$) for any simple ${\rm
C^*}$-algebra $A$ in the corresponding class of generalized tracial
approximation ${\rm C^*}$-algebras.
