On my 1950s vintage Ithaca Model 37 16 gauge it is on the right side of the end of the receiver where the magazine tube and barrel go into the receiver, and also on the barrel adjacent to the number on the receiver. The numbers should be the same on both the barrel and receiver unless the barrel is not the original barrel, at least on mine they are.
The Model 37 was mfg in 1969.
The serial number indicates Ithaca manufactured your Model 37 in 1956.
Call Ithaca at 1 877 648 4222 and they will tell you.
1969
The serial number indicates your Model 37 was manufactured in late 1969.
look on the Side of the recevier there should be a Serial Number on the older Ithaca Model 37 the serial Number was stamped on the receiver by the magazine tune and Barrel . also if you take the barrel off on the bottom of the barrel you'll find the matching serial Number. then look up WWW.SURPLUSRIFLEFORUM.COM in their search engine type in ITHACA 37 DOM match your shotguns Serial number up with their chart .
1947
Made in 1939
Made in 1963
1957
1947
1970