  For two meromorphic functions $ f $ and $ g $, the equation $ f^m+g^m=1 $ can
be regarded as Fermat-type equations. Using Nevanlinna theory for meromorphic
functions in several complex variables, the main purpose of this paper is to
investigate the properties of the transcendental entire solutions of
Fermat-type difference and partial differential-difference equations in $
\mathbb{C}^n $. In addition, we find the precise form of the transcendental
entire solutions in $ \mathbb{C}^2 $ with finite order of the Fermat-type
partial differential-difference equation $$\left(\frac{\partial
f(z_1,z_2)}{\partial z_1}\right)^2+(f(z_1+c_1,z_2+c_2)-f(z_1,z_2))^2=1$$ and
$$f^2(z_1,z_2)+P^2(z_1,z_2)\left(\frac{\partial f(z_1+c_1,z_2+c_2)}{\partial
z_1}-\frac{\partial f(z_1,z_2)}{\partial z_1}\right)^2=1,$$ where $P(z_1,z_2)$
is a polynomial in $\mathbb{C}^2$. Moreover, one of the main results of the
paper significantly improved the result of Xu and Cao [Mediterr. J. Math.
(2018) 15:227 , 1-14 and Mediterr. J. Math. (2020) 17:8, 1-4].
