  We introduce and discuss notions of regularity and flexibility for Lagrangian
manifolds with Legendrian boundary in Weinstein domains. There is a surprising
abundance of flexible Lagrangians. In turn, this leads to new constructions of
Legendrians submanifolds and Weinstein manifolds. For instance, many closed
$n$-manifolds of dimension $n>2$ can be realized as exact Lagrangian
submanifolds of $T^*S^n$ with possibly exotic Weinstein symplectic structures.
These Weinstein structures on $T^* S^n$, infinitely many of which are distinct,
are formed by a single handle attachment to the standard $2n$-ball along the
Legendrian boundaries of flexible Lagrangians. We also formulate a number of
open problems.
