The celebrated Margulis Lemma states that there exists a universal constant $c_n$ such that if all the generators of a discrete group of isometries of hyperbolic n-space displace some point with distance smaller than $c_n$, then the group is virtually nilpotent. In this talk, I will discuss a displacement constraint theorem in the spirit of Margulis Lemma for hyperbolic surfaces which can be applied to study quantitative geometry of hyperbolic surfaces such as the two-dimensional Margulis constant, lengths of loops on hyperbolic surfaces and generalized collar lemma.