  In this paper we study the following nonlinear Schr\"{o}dinger equation with
magnetic field \[ \Big(\frac{\varepsilon}{i}\nabla-A(x)\Big)^{2}u+V(x)u=f(|
u|^{2})u,\quad x\in\mathbb{R}^{2}, \] where $\varepsilon>0$ is a parameter,
$V:\mathbb{R}^{2}\rightarrow \mathbb{R}$ and $A: \mathbb{R}^{2}\rightarrow
\mathbb{R}^{2}$ are continuous potentials and $f:\mathbb{R}\rightarrow
\mathbb{R}$ has exponential critical growth. Under a local assumption on the
potential $V$, by variational methods, penalization technique, and
Ljusternick-Schnirelmann theory, we prove multiplicity and concentration of
solutions for $\varepsilon$ small.
