  We introduce two natural notions of multivariable Aluthge transforms (toral
and spherical), and study their basic properties. In the case of 2-variable
weighted shifts, we first prove that the toral Aluthge transform does not
preserve (joint) hyponormality, in sharp contrast with the 1-variable case.
Second, we identify a large class of 2-variable weighted shifts for which
hyponormality is preserved under both transforms. Third, we consider whether
these Aluthge transforms are norm-continuous. Fourth, we study how the Taylor
and Taylor essential spectra of 2-variable weighted shifts behave under the
toral and spherical Aluthge transforms; as a special case, we consider the
Aluthge transforms of the Drury-Arveson 2-shift. Finally, we briefly discuss
the class of spherically quasinormal 2-variable weighted shifts, which are the
fixed points for the spherical Aluthge transform.
