  Let $R$ be a commutative ring with non-zero identity and $M$ be a unitary
$R$-module. The goal of this paper is to extend the concept of 1-absorbing
primary ideals to 1-absorbing primary submodules. A proper submodule $N$ of $M$
is said to be a 1-absorbing primary submodule if whenever non-unit elements
$a,b\in R$ and $m\in M$ with $abm\in N$, then either $ab\in(N:_{R}M)$ or $m\in
M-rad(N).$ Various properties and chacterizations of this class of submodules
are considered. Moreover, 1-absorbing primary avoidance theorem is proved.
