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Best answer as chosen by Asker / Ratings
1. There are caves with some of the most rare materia in the game around the world map and to get to them you will need to do the chocobo sidequest and get the special chocobos heres what you need to do to get them.
-get stables to put the chocobos that you catch from choco Billy in the chocobo ranch.
-you will need greens to catch the chocobo and to raise their stats sylkas greens are the best and you can get them from Chocobo Sage on the nothern continent but you need to have the airship to get there.
-you will also need 2 types of nut, the carob nut which can be stolen from a dinosaur type enemy in the area near Chocobo Sage's house and the Zeio nut which can be stolen from goblins on goblin island which are located northwest of Midgar.
-you will need the chocobo lure materia which can be bought from Choco Billy.
-you will need a good and a great chocobo one must be male and the other female, you can catch good chocobos from the tracks near Gold Saucer and you can catch great chocobos from the tracks near Mideel, you will also need to catch a wonderful chocobo from the tracks near Icicle Inn.
-feeding your chocobos greens raises there stats which increases your chances of getting a special chocobo and racing them at Gold Saucer increases their class which also increases you chances of getting a special chocobo so do this before you mate them also once you mate them it will be a while before you can do it again.
Blue or Green chocobo:
To get a blue or green chocobo you need to mate the good and great chocobo with a carob nut the chocobo you get is random and if you don't get one keep trying until you do. One of them must be male and one must be female as you will need to mate them too. The blue chocobo can cross rivers and the green can go over mountains.
Black chocobo:
To get a black chocobo you need to mate the green and blue chocobos with a carob nut. The black chocobo has both the abilities of the blue and green chocobo.
Gold chocobo:
This is the best chocobo which has the abilities of the blue and green and can also cross oceans.

2. To get the Knight Of The Round you need to use the Gold chocobo. You will find island where they are in the Northeast corner of the map.

3. The Mystile can be bought in the Bone Village, the Imperial Guard is dropped by the Grand Horn just lose the battle in Fort Condor to fight him and can be found as a treasure inside North Crater and the Zeidrich can be stolen from the Turks during the fight in the Rocket Town and the Sunken Gelnika.

4. To get the underwater materia you need to morph the ghost ship found in the Junon Underwater reactor in the tube area, you will get a map which you need to give to the guy in Kalm to get the underwater materia.