  We survey known solutions to the infinite extendibility problem for
(necessarily exchangeable) probability laws on $\mathbb{R}^d$, which is: Can a
given random vector $\vec{X} = (X_1,\ldots,X_d)$ be represented in distribution
as the first $d$ members of an infinite exchangeable sequence of random
variables? This is the case if and only if $\vec{X}$ has a stochastic
representation that is "conditionally iid" according to the seminal de
Finetti's Theorem. Of particular interest are cases in which the original
motivation behind the model $\vec{X}$ is not one of conditional independence.
After an introduction and some general theory, the survey covers the
traditional cases when $\vec{X}$ takes values in $\{0,1\}^d$ has a spherical
law, a law with $\ell_1$-norm symmetric survival function, or a law with
$\ell_{\infty}$-norm symmetric density. The solutions in all These cases
constitute analytical characterizations of mixtures of iid sequences drawn from
popular, one-parametric probability laws on $\mathbb{R}$, like the Bernoulli,
the normal, the exponential, or the uniform distribution. The survey further
covers the less traditional cases when $\vec{X}$ has a Marshall-Olkin
distribution, a multivariate wide-sense geometric distribution, a multivariate
extreme-value distribution, or is defined as a certain exogenous shock model
including the special case when its components are samples from a Dirichlet
prior. The solutions in these cases correspond to iid sequences drawn from
random distribution functions defined in terms of popular families of
non-decreasing stochastic processes, like a L\'evy subordinator, a random walk,
a process that is strongly infinitely divisible with respect to time, or an
additive process. The survey finishes with a list of potentially interesting
open problems.
