  In the present work we analyse $\mathcal{N}=(2,2)$ supersymmetric Yang-Mills
(SYM) theory in two dimensions by means of lattice simulations. The theory
arises as dimensional reduction of $\mathcal{N}=1$ SYM theory in four
dimensions. As in other gauge theories with extended supersymmetry, the
classical scalar potential has flat directions which may destabilize numerical
simulations. In addition, the fermion determinant need not be positive and this
sign-problem may cause further problems in a stochastic treatment. We
demonstrate that $\mathcal{N}=(2,2)$ super Yang-Mills theory has actually no
sign problem and that the flat directions are lifted and thus stabilized by
quantum corrections. Only the bare mass of the scalars experience a finite
additive renormalization in this finite theory. On various lattices with
different lattice constants we determine the scalar masses and hopping
parameters for which the supersymmetry violating terms are minimal. By studying
four Ward identities and by monitoring the $\pi$-mass we show that
supersymmetry is indeed restored in the continuum limit. In the second part we
calculate the masses of the low-lying bound states. We find that in the
infinite-volume and supersymmetric continuum limit the Veneziano-Yankielowicz
super-multiplet becomes massless and the Farrar-Gabadadze-Schwetz
super-multiplet decouples from the theory. In addition, we estimate the masses
of the excited mesons in the Veneziano-Yankielowicz multiplet. We observe that
the gluino-glueballs have comparable masses to the excited mesons.
