The Dirac operator is a first order partial differential operator acting in the space of vector valued functions $L^2(\mathbb{R}^3; \mathbb{C}^4)$. It is the relativistic counterpart of the Schr\"odinger operator and hence, it appears in many applications in quantum mechanics. In this talk I will discuss the Dirac operator with an electrostatic $\delta$-shell interaction which is formally given by $A_\eta := A_0 + \eta \delta_\Sigma I_4$, where $A_0$ is the free Dirac operator, $I_4$ is the $4 \times 4$ identity matrix, $\eta \in \mathbb{R}$ and $\Sigma$ is the boundary of a smooth domain. After establishing the self-adjointness of $A_\eta$ it turns out that some of the spectral properties of $A_\eta$ are of a different nature, if $\eta = 2 c$, where $c$ denotes the speed of light, or $\eta \neq 2 c$. In the latter case functions in $\mathrm{dom}\, A_\eta$ are $H^1$-smooth in $\mathbb{R}^3 \setminus \Sigma$, the discrete spectrum is finite and the nonrelativistic limit of $A_\eta$, as the speed of light $c$ tends to $\infty$, can be computed. In the case $\eta = 2 c$ these properties either differ from those for $\eta \neq 2 c$ or they are still unknown. This talk is based on joint works with J. Behrndt, P. Exner, and V. Lotoreichik.