  We discuss the existence and characterization of quasi-stationary
distributions and Yaglom limits of self-similar Markov processes that reach 0
in finite time. By Yaglom limit, we mean the existence of a deterministic
function $g$ and a non-trivial probability measure $\nu$ such that the process
rescaled by $g$ and conditioned on non-extinction converges in distribution
towards $\nu$. If the study of quasi-stationary distributions is easy and
follows mainly from a previous result by Bertoin and Yor \cite{BYFacExp} and
Berg \cite{bergI}, that of Yaglom limits is more challenging. We will see that
a Yaglom limit exits if and only if the extinction time at 0 of the process is
in the domain of attraction of an extreme law and we will then treat separately
three cases, according whether the extinction time is in the domain of
attraction of a Gumbel law, a Weibull law or a Fr\'echet law. In each of these
cases, necessary and sufficient conditions on the parameters of the underlying
L\'evy process are given for the extinction time to be in the required domain
of attraction. The limit of the process conditioned to be positive is then
characterized by a multiplicative equation which is connected to a
factorization of the exponential distribution in the Gumbel case, a
factorization of a Beta distribution in the Weibull case and a factorization of
a Pareto distribution in the Fr\'echet case. This approach relies partly on
results on the tail distribution of the extinction time, which is known to be
distributed as the exponential integral of a L\'evy process. In that aim, new
results on such tail distributions are given, which may be of independent
interest. Last, we present applications of the Fr\'echet case to a family of
Ornstein-Uhlenbeck processes.
