Modular Invariance

I am confused about the significance of modular invariance in Conformal
Field Theory. What do statements like "a good conformal theory, has to
be modular invariant", really mean?

I understand that any CFT on the torus has to be modular invariant, to
respect the discrete symmetries of the torus.
If the conformal field theory is defined on the plane, does it still
need to be modular invariant to be a "good" theory? Why?

Thanks for any help,
Marvin

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