\emph{Coiterative functions} can be explained categorically as
\emph{final coalgebraic morphisms}, once coinductive types are viewed
as final coalgebras. However, the coiteration schema which arises in
this way is too rigid to accommodate directly many inte\-re\-sting
classes of circular specifications.  In this paper, building on the
notion of \emph{$T$-coiteration} introduced by the third author and
capitalizing on recent work on \emph{bialgebras} by Turi-Plotkin and
Bartels, we introduce and illustrate various ge\-ne\-ra\-li\-zed
coiteration patterns.  First we show that, by choosing the appropriate
monad $T$, $T$-coiteration captures naturally a wide range of
coiteration schemata, such as the duals of \emph{primitive recursion}
and \emph{course-of-value} iteration, and \emph{mutual}
coiteration. Then we show that, in the more structured categorical
setting of bialgebras, $T$-coiteration captures \emph{guarded}
coiterations schemata, i.e. specifications where recursive calls
appear guarded by predefined algebraic operations.
\\ {\bf Keywords:} 
coinductive datatype, categorical semantics, coalgebra, bialgebra, coiteration
schema, guarded specification.