  We introduce and study a "desingularization" of a Lie groupoid $G$ along an
"$A(G)$-tame" submanifold $L$ of the space of units $M$. An $A(G)$-tame
submanifold $L \subset M$ is one that has, by definition, a tubular
neighborhood on which $A(G)$ becomes a thick pull-back Lie algebroid. The
construction of the desingularization $[[G:L]]$ of $G$ along $L$ is based on a
canonical fibered pull-back groupoid structure result for $G$ in a neighborhood
of the tame $A(G)$-submanifold $L \subset M$. This local structure result is
obtained by integrating a certain groupoid morphism, using results of Moerdijk
and Mrcun (Amer. J. Math. 2002). Locally, the desingularization $[[G:L]]$ is
defined using a construction of Debord and Skandalis (Advances in Math., 2014).
The space of units of the desingularization $[[G:L]]$ is $[M:L]$, the blow up
of $M$ along $L$. The space of units and the desingularization groupoid
$[[G:L]]$ are constructed using a gluing construction of Gualtieri and Li (IMRN
2014). We provide an explicit description of the structure of the
desingularized groupoid and we identify its Lie algebroid, which is important
in analysis applications. We also discuss a variant of our construction that is
useful for analysis on asymptotically hyperbolic manifolds. We conclude with an
example relating our constructions to the so called "edge pseudodifferential
calculus." The paper is written such that it also provides an introduction to
Lie groupoids designed for applications to analysis on singular spaces.
