Certain classes of supersymmetric Yang-Mills (SYM) theories including the well known N=4 SYM, that takes part in the AdS/CFT correspondence, can be formulated on a Euclidean spacetime lattice using the techniques of exact lattice supersymmetry. In this talk I will provide the constructions of such theories and describe how great ideas such as topological field theories, Dirac-Kahler fermions, geometric discretization all come together to create lattice SYM theories that are exact-supersymmetric, gauge-invariant, local and doubler-free on the lattice. Then I move on to discuss the recent lattice constructions of specific classes SYM theories in two dimensions, with matter in various representations.