  For $g\geq 2$, let $\mathrm{Mod}(S_g)$ be the mapping class group of the
closed orientable surface $S_g$ of genus $g$. In this paper, we obtain
necessary and sufficient conditions under which a given pseudo-periodic mapping
class can be a root of another up to conjugacy. Using this characterization,
the canonical decomposition of (non-periodic) mapping classes, and some known
algorithms, we give an algorithm for determining the conjugacy classes of roots
of arbitrary mapping classes. Furthermore, we derive realizable bounds on the
degrees of roots of pseudo-periodic mapping classes in $\mathrm{Mod}(S_g)$, the
Torelli group, the level-$m$ subgroup of $\mathrm{Mod}(S_g)$, and the
commutator subgroup of $\mathrm{Mod}(S_2)$. In particular, we show that the
highest possible (realizable) degree of a root of a pseudo-periodic mapping
class $F$ is $3q(F)(g+1)(g+2)$, where $q(F)$ is a unique positive integer
associated with the conjugacy class of $F$. Moreover, this bound is realized by
a root of a power of a Dehn twist about a separating curve of genus $[g/2]$ in
$S_g$, where $g\equiv 0,9 \pmod{12}$. Finally, for $g\geq 3$, we show that any
pseudo-periodic mapping class having a nontrivial periodic component that is
not the hyperelliptic involution, normally generates $\mathrm{Mod}(S_g)$.
Consequently, we establish that $\mathrm{Mod}(S_g)$ is normally generated by a
root of bounding pair map or a root of a nontrivial power of a Dehn twist.

Geometric Topology