You know all about accidentals? Now it is time to learn quarter tones.
Quarter tones are used in contemporary music or to adapt eastern music to western music notation.
Notation of quarter tones is still not completely standardized, so you can find many symbols for the same signification.
Half flat
Half flat lowers the pitch by a quarter tone.
There are several symbols for half flats, here they are:
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Half sharp
Half sharp raises the pitch by a quarter tone.
There are several symbols for half sharps, here they are:
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Three-quarter-tone flat (or flat and a half)
Three-quarter-tone flat (or flat and a half) lowers the pitch by a 3 quarters tone.
There are several symbols for three-quarter-tone flat, here they are:
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Sharp-and-a-half (or three-quarter-tone sharp)
Sharp-and-a-half (or three-quarter-tone sharp) raises the pitch by 3 quarters tone.
There are several symbols for three Sharp-and-a-half, here they are:
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Summary of quarter tones
Here are all quarter tones notations:
Half flats | Half sharps | Sharp-and-a-half | Sharp-and-a-half | ||||||||
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Quarter tone scale
Below you can hear a quarter tone scale.

Where do you find quarter tones?
Quarter tones are used in contemporary music, and in some oriental scales such as Arabic maqams. The term maqam is used to refer to Arabic modes (Arabic scales), and a lot of maqam uses quarter tones.
Example with the Bayati maqam (you will notice the half flat on the E note):

External links :
1 - https://en.wikipedia.org/wiki/Quarter_tone
2 - https://musicterms.artopium.com/q/Quartertone.htm