We show that the Dirichlet to Neumann map associated to the quasilinear isotropic elliptic equation \(\nabla \cdot \gamma(x, u) \nabla u= 0\) determines uniquely the scalar coefficient \(\gamma(x, z)\), where \((x, z)\in \Omega\times \mathbb{R}\), \(\Omega\subset \mathbb{R}^n\) and \(n\geq 2\). This result generalizes a well-known global uniqueness theorem for an inverse boundary value problem for the linear isotropic elliptic equation \(\nabla\cdot \gamma(x) \nabla u= 0\) to quasilinear isotropic elliptic equations. We also consider the case of quasilinear anisotropic elliptic equations, where \(\gamma(x,z)\) is replaced by a positive definite matrix function \(A(x, z)\). We study an example in which we show that the Dirichlet to Neumann map determines the matrix coefficient \(A(x, z)\) modulo the group of diffeomorphisms which are the identity on the boundary of \(\Omega\).