In this paper, the authors firstly introduce the history of the study of differential equations with nonstandard growth conditions as well as corresponding function spaces with variable exponents of nonlinearity and secondly state some recent results of solutions to nonlinear elliptic problems. Since nonlinear PDEs with variable exponent arise from modeling the real problems such as image processing and electro-rheological fluids, this issue has caused wide public concern. The authors study such problem and obtain some results of nonlinear elliptic and parabolic equations with nonstandard growth conditions by $\textrm{Rothe}$'s method, $\textrm{Galerkin}$'s method, Variational method and fixed point theorems.