The exact name of the unit is Luftlande Sturm Regiment, commonly written as LL Sturm Regiment. Luftlande meaning airlanding as these guys were all dual trained (until Kreta) for jumping and glider ops.
The unit began in 1939 with members of 1/FJR1 and the Pioneer zug (engineer platoon) from II/FJR under the command of Hauptmann Koch. The unit called Sturmabteilung (assault detachment) Koch and prepared for the assault on Eben Emael. The unit assaulted Eben Emael in Belgium on 10 May 1940. After the campaign in the west in 1940, the 7th Flieger Division (the FJ division) was enlarged and consisted of 4 parachute regiments;FJR1, FJR2, FJR3 and the LL Sturm Regiment. The LL Sturm Regiment was formed and consisted of 4 battalions. Both the LLSturm Regiment and the 7th Flieger division were fully formed by the end of 1940.
On 20 May 1941 the entire LL Sturm Regiment assaulted Kreta at the airfield at Maleme. They captured the airfield and hill 107, which then allowed the 5th Gebirgsjaeger to be flown in.
After Kreta they went back to Germany to reform and by the end of 1941 all 4 battalions were sent to different locations on the Russian Front. During the middle parts of 1942 the regiment was pulled from the Russian Front. Parts of the regiment formed II/FJR6 and FJR5.
FJR5 went to Tunisia during the end of 1942 and fought the Allis where it surrendered, with few getting out of Africa.
The rest of the regiment (and survivors of FJR5) formed FJR12 which became part of the 4th FJD in late 1943. FJR12 was refered to as the Sturm Regiment till the end of the war.
They were deployed to Italy during the fall of 1943 and fought at Salerno, Anzio and the south of Rome. The unit was pulled up to the north after FJD1 pulled out of Cassino and fought the rest of the war in northern Italy