Permutation polynomials over finite fields have been an important subject of study for a long time and have wide applications in coding theory, cryptography and sequence designs. However, only some specific classes of permutation polynomials have been described in the literature so far. In this paper, based on the knowledge of finite fields, such as the properties of the trace function, we propose two classes of permutation polynomials having the form $(x^{2^{i}}+\eta x+\delta)^{s}+x$ and two classes of permutation trinomials having the form $x^{r}+\delta x^s+\delta^{t}x$ over the finite field $\mathbb{F}_{2^n}$. The permutation polynomials of the first form are studied along with the work of Yuan and Ding, and the method studying the permutation polynomials of the second form relies a sufficient condition for a trinomial having no root, which is established in this paper. There are rare known classes of permutation trinomials over $\mathbb{F}_{2^n}$ in the literature, and most of them are of the simple form, i.e., all nonzero coefficients equal to the identity. The permutation trinomials presented in this paper have the coefficients which can take any nonzero elements in $\mathbb{F}_{2^n}$.