  We study truncated objects using elementary methods. Concretely, we use
universes and the resulting natural number object to define internal truncation
levels and prove they behave similar to standard truncated objects. Moreover,
we take an elementary approach to localizations, giving various equivalent
conditions that characterize localizations and constructing a localization out
of a sub-universe of local objects via an internal right Kan extension. We then
use this general approach, as well as an inductive approach, to construct
truncation functors. We use the resulting truncation functors to prove
classical results about truncations, such as Blakers-Massey theorem, in the
elementary setting. We also give examples of non-presentable $(\infty,
1)$-categories where the elementary approach can be used to define and compute
truncations. Finally, we turn around and use truncations to study elementary
$(\infty, 1)$-toposes and show how they can help us better understand subobject
classifiers and universes
