Rotating　machinery　often　works　under　time-varying　speeds,　and　nonstationary　conditions　and　harsh　environments　make　its　key　parts,　such　as　rolling　bearings　and　gears,　prone　to　faults.　Therefore,　a　number　of　fault　diagnosis　methods,　including　nonstationary　signal　processing　methods　and　data-driven　methods,　have　been　developed.　This　article　presents　a　comprehensive　review　of　the　fault　diagnosis　of　rotating　machinery　under　time-varying　speeds　proposed　mainly　during　the　last　five　years.　First,　spectrum　analysis-based　methods,　including　order　tracking,　cyclic　spectrum　correlation,　and　generalized　demodulation,　are　reviewed.　Second,　the　time-frequency　analysis　(TFA)　methods　in　machinery　fault　diagnosis　are　divided　into　postprocessing　methods　and　chirplet　transform-based　methods　and　are　reviewed.　Then,　the　artificial　feature　extraction-　and　deep　learning-enabled　intelligent　diagnosis　methods　proposed　specifically　for　time-varying　speed　conditions　are　reviewed.　Finally,　the　research　prospects　are　discussed.　We　not　only　review　the　relevant　state-of-the-art　methods　and　analyze　how　they　overcome　the　problems　caused　by　speed　fluctuations　but　also　discuss　their　advantages　and　disadvantages　and　the　challenges　that　will　be　encountered　when　applying　them　to　industrial　applications.　This　article　is　expected　to　provide　new　graduate　students,　institutions,　and　companies　with　a　preliminary　understanding　of　the　methods　on　this　topic.