This talk concerns a joint work with Paolo Bellingeri and Anne-Laure Thiel. Starting from the observation that the standard presentation of a virtual braid group mixes the presentations of the corresponding braid group and the corresponding symmetric group together with the action of the symmetric group on its root system, we define a virtual Artin group ${\rm VA}[\Gamma]$ with a presentation that mixes the standard presentations of the Artin group $A[\Gamma]$ and of the Coxeter group $W[\Gamma]$ together with the action of $W[\Gamma]$ on its root system. By definition we have two epimorphisms $\pi_K:{\rm VA}[\Gamma]\to W[\Gamma]$ and $\pi_P:{\rm VA}[\Gamma]\to W[\Gamma]$ whose kernels are denoted by ${\rm KVA}[\Gamma]$ and ${\rm PVA}[\Gamma]$, respectively. In this talk we will focus on ${\rm KVA}[\Gamma]$. We will show that this group is an Artin group whose standard generating set is in one-to-one correspondence with the root system of $W[\Gamma]$. Afterwards, we use this presentation to show that the center of ${\rm VA}[\Gamma]$ is always trivial, and to show that ${\rm VA}[\Gamma]$ has a solvable word problem and finite virtual cohomological dimension when $\Gamma$ is of spherical type or of affine type.