  Extriangulated categories were introduced by Nakaoka and Palu by extracting
the similarities between exact categories and triangulated categories. A notion
of mutation of subcategories in an extriangulated category is defined in this
article. Let $\cal A$ be an extension closed subcategory of an extriangulated
category $\cal C$. Then the quotient category $\cal M:=\cal{A}/\cal{X}$ carries
naturally a triangulated structure whenever $(\cal A,\cal A)$ forms an $\cal
X$-mutation pair. This result unifies many previous constructions of
triangulated quotient categories, and using it gives a classification of thick
triangulated subcategories of pretriangulated category $\cal{C}/\cal{X}$, where
$\cal X$ is functorially finite in $\cal C$. When $\cal C$ has Auslander-Reiten
translation $\tau$, we prove that for a functorially finite subcategory $\cal
X$ of $\cal C$ containing projectives and injectives, $\cal{C}/\cal{X}$ is a
triangulated category if and only if $(\cal C,\cal C)$ is $\cal X-$mutation if
and only if $\tau \underline{\cal X}=\bar{\cal X}.$ This generalizes a result
by J{\o}rgensen who proved the equivalence between the first and the third
conditions for triangulated categories. Furthermore, we show that for such a
subcategory $\cal X$ of the extriangulated category $\cal C$, $\cal C$ admits a
new extriangulated structure such that $\cal C$ is a Frobenius extriangulated
category. Applications to exact categories and triangulated categories are
given. From the applications we present examples that extriangulated categories
are neither exact categories nor triangulated categories.
