  We introduce twisted relative Cuntz-Krieger algebras associated to finitely
aligned higher-rank graphs and give a comprehensive treatment of their
fundamental structural properties. We establish versions of the usual
uniqueness theorems and the classification of gauge-invariant ideals. We show
that all twisted relative Cuntz-Krieger algebras associated to finitely aligned
higher-rank graphs are nuclear and satisfy the UCT, and that for twists that
lift to real-valued cocycles, the K-theory of a twisted relative Cuntz-Krieger
algebra is independent of the twist. In the final section, we identify a
sufficient condition for simplicity of twisted Cuntz-Krieger algebras
associated to higher-rank graphs which are not aperiodic. Our results indicate
that this question is significantly more complicated than in the untwisted
setting.
