The functional equation $$f^{m}+g^{m}=1$$ fm+gm=1 can be regarded as the Fermat-type equations over function fields. In this paper, we investigate the entire and meromorphic solutions of the Fermat-type functional equations such as partial differential-difference equation $$\left( \frac{\partial f(z_{1}, z_{2})}{\partial z_{1}}\right) ^{n}+f^{m}(z_{1}+c_{1}, z_{2}+c_{2})=1$$ ∂f(z1,z2)∂z1n+fm(z1+c1,z2+c2)=1 in $$\mathbb {C}^{2}$$ C2 and partial difference equation $$f^{m}(z_{1}, \ldots , z_{n})+f^{m}(z_{1}+c_{1}, \ldots , z_{n}+c_{n})=1$$ fm(z1,…,zn)+fm(z1+c1,…,zn+cn)=1 in $$\mathbb {C}^{n}$$ Cn by making use of Nevanlinna theory for meromorphic functions in several complex variables.