The 2D eigenvalue problem (2dEVP) is a class of the double eigenvalue problems first studied by Blum and Chang in 1970s. The 2dEVP seeks real scalars $\lambda, \mu$,

and a corresponding vector $x$ satisfying the following equations

\begin{align*}

Ax & = \lambda x + \mu Cx,\\

x^H C x &=0, \\

x^H x &=1,

\end{align*}

where $A$ and $C$ are Hermitian and $C$ is indefinite. We show the connections between 2dEVP with well-known numerical linear algebra and optimization problems such as quadratic programming, the distance to instability and $H_{\infty}$-norm. We will discuss (1) fundamental properties of 2dEVP including well-posedness, types and regularity, (2) backward error analysis and numerical algorithms.