  Let $G$ be a linear algebraic group over an infinite field $k$. Loosely
speaking, a $G$-torsor over $k$-variety is said to be versal if it specializes
to every $G$-torsor over any $k$-field. The existence of versal torsors is
well-known. We show that there exist $G$-torsors that admit even stronger
versality properties. For example, for every $d\in\mathbb{N}$, there exists a
$G$-torsor over a smooth quasi-projective $k$-scheme that specializes to every
torsor over a quasi-projective $k$-scheme after removing some codimension-$d$
closed subset from the latter. Moreover, such specializations are abundant in a
well-defined sense. Similar results hold if we replace $k$ with an arbitrary
base-scheme. In the course of the proof we show that every globally generated
rank-$n$ vector bundle over a $d$-dimensional $k$-scheme of finite type can be
generated by $n+d$ global sections.
  When $G$ can be embedded in a group scheme of unipotent upper-triangular
matrices, we further show that there exist $G$-torsors specializing to every
$G$-torsor over any affine $k$-scheme. We show that the converse holds when
$\operatorname{char} k=0$.
  We apply our highly versal torsors to show that, for fixed
$m,n\in\mathbb{N}$, the symbol length of any degree-$m$ period-$n$ Azumaya
algebra over any local $\mathbb{Z}[\frac{1}{n},e^{2\pi i/n}]$-ring is uniformly
bounded. A similar statement holds in the semilocal case, but under mild
restrictions on the base ring.
