In 2005 the author of the present paper introduced the theory of \(G\)-normal distributions, \(G\)-expectations (a kind of nonlinear expectations), the one-dimensional \(G\)-Brownian motion and the associated stochastic calculus [Chin. Ann. Math., Ser. B 26, No. 2, 159–184 (2005; Zbl 1077.60045)]. Unlike the classical normal distribution \({\mathcal N}(0,\sigma^2)\), characterized by the linear heat equation \(\partial_tu(t,x)=1/2\cdot\sigma^2 \partial_{xx}^2u(t,x)\), the \(G\)-normal distribution \({\mathcal N}(0,[\sigma^2_1\sigma^2_2])\) with \(G(a):=\sigma^2_2 a^+-\sigma^2_1a^-,\) \(a\in\mathbb R\), is associated to the nonlinear heat equation \(\partial_tu(t,x)=1/2\cdot G(\partial_{xx}^2u(t,x)).\) This theory has been motivated by the earlier studies of \(g\)-expectations defined with the help of a backward stochastic differential equation [S. Peng, BSDE and stochastic optimizations, in: J. Yan, S. Peng, S. Fang, L. Wu (eds.), Topics in Stochastic Analysis, Science Press, Beijing (1997)] and by describing coherent and dynamic risk measures. In the present paper the author extends his former studies to the multi-dimensional \(G\)-Brownian motion: after having introduced multi-dimensional \(G\)-normal distributions via the nonlinear heat equation he defines the nonlinear \(G\)-expectation \(\widehat{E}\) on \(C_0(\mathbb R_+,\mathbb R^n)\) under which the coordinate process \(B\) is a process whose future increments are independent of the past ones and \(G\)-normally distributed. Such a process is called \(G\)-Brownian motion. The author then develops the related stochastic calculus: He introduces, in particular, the stochastic integral with respect to a \(G\)-Brownian motion, establishes an Itô formula and studies stochastic differential equations driven by a \(G\)-Brownian motion. The theory developed by the author has to be regarded as analog of the classical stochastic Itô calculus under the nonlinear expectation \(\widehat{E}\).