A stochastic process $\lbrace X(t)\mid t\in T\rbrace$ is said to be a Gaussian process if all of the members of its f.f.d. (family of finite dimensional distributions) are joint normal distributions. In other words, for any positive integer $n<\infty$ and any $t_1,\ldots,t_n\in T$ the joint distribution of random variables $X(t_1),\ldots,X(t_n)$ is jointly normal.

As an example, any Wiener process is Gaussian.

Remark. Sometimes, a Gaussian process is known as a Gaussian random field if $T$ is a subset, usually an embedded manifold, of $\mathbb{R}^m$ with $m>1$