Linear Phase Response
In some (communication) systems a linear phase response is desired because of the group delay (group delay is nothing else than the negative SLOPE of the phase function).
Hence, a linear phase gives a constant group delay.
By the way: A "constant" phase is possible for a pure resistive circuit only (without frequency dependence). 

However, there are many applications (filters) which do not require linear phase functions because group delay is not of primary concern.