In this talk I will describe our work on the simulation of the Schwinger model (i.e. d=1+1 QED) with matrix product states (MPS). I will discuss some systematic aspects of our approach like the truncation of the local infinite bosonic gauge field Hilbert space, or the incorporation of local gauge invariance into the MPS anzats. Furthermore, I will go through some of our results: the simulation of the particle excitations ('mesons' of confined electron/positron pairs), of string breaking for heavy probe charges and last but not least of the real-time evolution that occurs from a background electric field quench (i.e. the full quantum Schwinger effect).
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