  We introduce a notion of Mathieu-Zhao subspaces of vertex algebras. Among
other things, we show that for a vertex algebra $V$ and its subspace $M$ that
contains $C_2(V)$, $M$ is a Mathieu-Zhao subspace of $V$ if and only if the
quotient space $M/C_2(V)$ is a Mathieu-Zhao subspace of a commutative
associative algebra $V/C_2(V)$. As a result, one can study the famous Jacobian
conjecture in terms of Mathieu-Zhao subspaces of vertex algebras. In addition,
for a $CFT$-type vertex operator algebra $V$ that satisfies the
$C_2$-cofiniteness condition, we classify all Mathieu-Zhao subspaces $M$ that
contain $C_2(V)$.
