Sergeant Major, military rank:
A Sergeant Major (abbreviated SGM or SgtMaj) in the United States Marine Corps and Army is the most senior enlisted person in the service, holding a pay grade of E-9. They serve as the chief administrative assistant of a headquarters unit of the Marine Corps or Army.
Most Sergeants Major in the Army are trained at the United States Army Sergeants Major Academy (or, abbreviated, the USASMA), which is part of the Noncommissioned Officer Education System (NCOES). The USAMSA, located in Fort Bliss, Texas, is designed to develop common leader training for all Advanced Noncommissioned Officer Courses, covering leadership, communications, training management, professional skills, and military studies.
The shoulder insignia of a Sergeant Major in either the Marine Corps or Army are quite similar. In the Marine Corps, three green upward-facing chevrons cap four green upward-facing shallow arches. A green star is centered in the area created by the arches and chevrons. This is mounted on a red background. In the Army, three yellow upward-facing arches are topped by three yellow upward-facing chevrons with a yellow five-pointed star in the center. This is mounted on an olive green background.
The rank of Sergeant Major is comparable to a Master Chief Petty Officer in the United States Navy or a Chief Master Sergeant or First Sergeant of the United States Air Force.
In the British Army, Sergeant Major describes two special titles: Company Sergeant Major (CSM) and Regimental Sergeant Major (RSM). These titles are given to the most senior Warrant Officers Class 2 (for CSM) or Warrant Officers Class 1 (for RSM) when they take a leadership position in a company or regiment rather than signifying a separate rank.
As the most senior enlisted person, Sergeants Major have an unusual amount of authority in the British Army. They are in responsible for the training and discipline of all enlisteds and NCOs in their regiment/company and report directly to the second-in-command.
Unlike their American counterparts, however, Sergeants Major in the British Army are considered warrant officers instead of noncommissioned officers, so their pay grades are WO1 (for a RSM) and WO2 (for a CSM). They are addressed as "sir" by their subordinates but are not entitled to a salute.
Company and Regimental Sergeant Majors do have insignia unique from that of warrant officers. A Company Sergeant Major is entitled to wear a crown badge at the bottom of their sleeve while Regimental Sergeant Majors wear a Royal Coat of Arms at the bottom of the sleeve.
The Australian and New Zealand armies have the same system for sergeant majors as the British.
Further information on sergeants major in other countries' armed forces will be added as I find it.