  Records among a sequence of iid random variables $X_1,X_2,\dotsc$ on the real
line have been investigated extensively over the past decades. A record is
defined as a random variable $X_n$ such that $X_n>\max(X_1,\dotsc,X_{n-1})$.
Trying to generalize this concept to the case of random vectors, or even
stochastic processes with continuous sample paths, the question arises how to
define records in higher dimensions. We introduce two different concepts: A
simple record is meant to be a stochastic process (or a random vector) $X_n$
that is larger than $ X_1,\dotsc, X_{n-1}$ in at least one component, whereas a
complete record has to be larger than its predecessors in all components. The
behavior of records is investigated. In particular, the probability that a
stochastic process $ X_n$ is a record as $n$ tends to infinity is studied,
assuming that the processes are in the max-domain of attraction of a max-stable
process. Furthermore, the distribution of $ X_n$, given that $ X_n$ is a record
is derived.
