  Let $x \in S^{n-1}$ be a unit eigenvector of an $n \times n$ random matrix.
This vector is delocalized if it is distributed roughly uniformly over the real
or complex sphere. This intuitive notion can be quantified in various ways. In
these lectures, we will concentrate on the no-gaps delocalization. This type of
delocalization means that with high probability, any non-negligible subset of
the support of $x$ carries a non-negligible mass. Proving the no-gaps
delocalization requires establishing small ball probability bounds for the
projections of random vector. Using Fourier transform, we will prove such
bounds in a simpler case of a random vector having independent coordinates of a
bounded density. This will allow us to derive the no-gaps delocalization for
matrices with random entries having a bounded density. In the last section, we
will discuss the applications of delocalization to the spectral properties of
Erd\H{o}s-R\'enyi random graphs.
