We investigate the distribution of zeros and shared values of the difference operator on meromorphic functions. In particular, we show that if f is a transcendental meromorphic function of finite order with a small number of poles, c is a non-zero complex constant such that ${\Delta }_c^{k}f\ne 0$ for n ≥ 2, and a is a small function with respect to f, then $fⁿ{\Delta }_c^{k}f$ equals a (≠ 0,∞) at infinitely many points. Uniqueness of difference polynomials with the same 1-points or fixed points is also proved.