  Let $\A$ be an algebra and $\sigma$ an automorphism of $\A$. A linear map $d$
of $\A$ is called a $\sigma$-derivation of $\A$ if $d(xy) = d(x)y +
\sigma(x)d(y)$, for all $x, y \in \A$. A bilinear map $D: \A \times \A \to \A$
is said to be a $\sigma$-biderivation of $\A$ if it is a $\sigma$-derivation in
each component. An additive map $\Theta$ of $\A$ is $\sigma$-commuting if it
satisfies $\Theta(x)x - \sigma(x)\Theta(x) = 0$, for all $x \in \A$. In this
paper, we introduce the notions of inner and extremal $\sigma$-biderivations
and of proper $\sigma$-commuting maps. One of our main results states that,
under certain assumptions, every $\sigma$-biderivation of a triangular algebras
is the sum of an extremal $\sigma$-biderivation and an inner
$\sigma$-biderivation. Sufficient conditions are provided on a triangular
algebra for all of its $\sigma$-biderivations (respectively, $\sigma$-commuting
maps) to be inner (respectively, proper). A precise description of
$\sigma$-commuting maps of triangular algebras is also given. A new class of
automorphisms of triangular algebras is introduced and precisely described. We
provide many classes of triangular algebras whose automorphisms can be
precisely described.
