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ASTM (American Society for Testing Materials) Specifications
These are basically materials specifications.  ASTMA A-403 (Factory Made Wrought Austenitic Steel Welding Fittings) lists the permissible raw materials from which fittings can be made.  In addition, it controls the method of manufacture, quality assurance and marking.
The corresponding ASME Spec is SA-403.  This is found in Section II Part A of the ASME Boiler and Pressure Vessel Code.  This is similar, but not identical to ASTM A-403.

ASME (American Society of Mechanical Engineers) Boilers and Pressure Vellel Code and MSS (Manufacturers Standardization Society) Standards
Both ASME B 16.9 entitled "Wrought Steel Butt Welding Fittings" and MSS SP-43 entitled "Wrought Stainless Steel Butt Welding Fittings" cover 1/2" through 24" and govern dimension and tolerance.  The difference is that ASME B 16.9 covers all wall thicknesses and requires fittings to be fully rated to the pressure temperature ratings of the matching pipe.
MSS SP-43 covers only light weight stainless steel fittings, and allows lower pressure temperature ratings than B 16.9.  See below for details:

Other supplementary ASME and MSS standards:

ASME B 16.25 - Butt welding end preparations
ASME B 16.28 - Butt welding short radius elbows and returns 1" - 24"
MSS SP 25 - Marking of each fitting
ASME Section II - Materials.  Part A - Ferrous; Part B - Non-Ferrous; Part C - Welding Materials; Part D - Properties
ASME Section III - Nuclear Power Plant Components.  For the manufacture of fittings where filler metal is added, a Certificate of Authorization (NPT Stamp) is required.
ASME Section V - Non-destructive examination
ASME Section VIII - Pressure vessels
ASME Section IX - Welding Qualifications

MSS SP-43 and ASME B16.9 for Schedules 5S and 10S Fittings
In response to the need of the process industries, SP-43 Specification was issued in 1950 by the Manufacturers Standardization Society.  At that time, stainless steel piping was used for its resistance to corrosion and to eliminate product contamination.  Pressure was seldom a factor.
Pressure ratings for 5S and 10S fittings are shown in MSS SP-43.  They reflect the emphasis on corrosion resistance rather than pressure.  These pressure ratings are less than the allowable pressure ratings that are applicable to the matching pipe.
Today there are many applications for schedules 5S and 10S piping at higher pressures than those permitted in SP-43.  To cover this, SP-43 was revised to include a statement, "For fittings of same pressure rating of matchine pipe, refer to ASME-B 16.9."
SP-43 and B 16.9 are similar dimensionally in that the fixed position for the welding ends with reference to the centerline of the fitting and the overall dimensions are identical.
ASME B 16.9 has the added provision that the pressure of the fitting shall at least equal the computed bursting pressure of seamless pipe of the same wall thickness and material.
In order to differentiate fittings made to these specifications, the prefix "CR" is shown before the alloy designated for SP-43 and the symbol "WP" is shown to indicate B 16.9 fittings.
A user who wants to obtain 5S and 10S to B 16.9 fittings must reference this Spec when ordering fittings to ASTM A-403.  If B 16.9 is not specified, the manufacturer may supply either SP-43 or B 16.9 at his option.