We will study moduli spaces of (abstract and embedded) tropical curves and surfaces. The computation of such moduli spaces is highly non-trivial and poses challenges which are interesting to master. Recent results show that moduli spaces of (planar) embedded tropical curves do not behave exactly as one would expect from the perspective of algebraic geometry. We investigate modifications and compactifications as means to overcome this inconsistency. A special focus is set on curves of genus two (in connection with Igusa invariants as appearing in number theory) and genus three.