  Experimental evidence demonstrating that anomalous localization of waves can
be induced in a controllable manner is reported. A microwave waveguide with
dielectric slabs randomly placed is used to confirm the presence of anomalous
localization. If the random spacing between slabs follows a distribution with a
power-law tail (L\'evy-type distribution), unconventional properties in the
microwave-transmission fluctuations take place revealing the presence of
anomalous localization. We study both theoretically and experimentally the
complete distribution of the transmission through random waveguides
characterized by $\alpha=1/2$ ("L\'evy waveguides") and $\alpha=3/4$, $\alpha$
being the exponent of the power-law tail of the L\'evy-type distribution. As we
show, the transmission distributions are determined by only two parameters,
both of them experimentally accessible. Effects of anomalous localization on
the transmission are compared with those from the standard Anderson
localization.
