  We provide several new $q$-congruences for truncated basic hypergeometric
series, mostly of arbitrary order. Our results include congruences modulo the
square or the cube of a cyclotomic polynomial, and in some instances,
parametric generalizations thereof. These are established by a variety of
techniques including polynomial argument, creative microscoping (a method
recently introduced by the first author in collaboration with Zudilin),
Andrews' multiseries generalization of the Watson transformation, and
induction. We also give a number of related conjectures including congruences
modulo the fourth power of a cyclotomic polynomial.
