  Let $\Gamma(S)$ be the pure mapping class group of a connected orientable
surface $S$ of negative Euler characteristic. For ${\mathscr C}$ a class of
finite groups, let $\hat{\pi}_1(S)^{\mathscr C}$ be the pro-${\mathscr C}$
completion of the fundamental group of $S$. The \emph{${\mathscr C}$-congruence
completion $\check{\Gamma}(S)^{\mathscr C}$ of $\Gamma(S)$} is the profinite
completion induced by the embedding
$\Gamma(S)\hookrightarrow{\operatorname{Out}}(\hat{\pi}_1(S)^{\mathscr C})$. In
this paper, we begin a systematic study of such completions for different
${\mathscr C}$. We show that the combinatorial structure of the profinite
groups $\check{\Gamma}(S)^{\mathscr C}$ closely resemble that of $\Gamma(S)$. A
fundamental question is how ${\mathscr C}$-congruence completions compare with
pro-${\mathscr C}$ completions. Even though, in general (e.g.\ for ${\mathscr
C}$ the class of finite solvable groups), $\check{\Gamma}(S)^{\mathscr C}$ is
not even virtually a pro-${\mathscr C}$ group, we show that, for ${\mathbb
Z}/2\in{\mathscr C}$, $g(S)\leq 2$ and $S$ open, there is a natural epimorphism
from the ${\mathscr C}$-congruence completion $\check{\Gamma}(S)(2)^{\mathscr
C}$ of the abelian level of order $2$ to its pro-${\mathscr C}$ completion
$\widehat{\Gamma}(S)(2)^{\mathscr C}$. In particular, this is an isomorphism
for the class of finite groups and for the class of $2$-groups. Moreover, in
these two cases, the result also holds for a closed surface.
