Abstract: Due to the introduction of Nevanlinna theory of meromorphic functions, the researches on the complex difference , complex q-difference, difference equations and q-difference equations are developing rapidly in recent decades. Some main results on these topics in recent decades are investigated in this paper. These results include the difference analogue of the logarithmic derivative; the difference analogue of Clunie lemma;the difference counterpart of Mohon'ko lemma; the properties on the zeros, fixed-points on complex differences and divided difference of meromorphic functions with small order; the properties on the value distribution of difference polynomials, the properties on the meromorphic solutions of difference Riccati and Painlev\'e equations; the results on q-differences and meromorphic solutions of q-difference equations.