In the light of great importance of curves and their frames in many different branches of science, especially differential geometry as well as geometric properties and the uses in various fields, we are interested here to study a special kind of curves called Smarandache curves in Lorentz 3-space. Then, we present some characterizations for these curves and calculate their Darboux invariants. Moreover, we classify TP, TU, PU and TPU-Smarandache curves of a spacelike curve according to the causal character of the vector, curve and surface used in the study. Besides, we give some of differential geometric properties and important relations between that curves. Finally, to demonstrate our theoretical results a computational example is given with graph.