Photons have mass, but no rest mass. (Or something like that.)

Momentum but no mass, if I recall my physics correctly (low certainty).

Wait, hang on, we have access to an appreciable-chunk of the world's knowledge at our fingertips...

https://en.wikipedia.org/wiki/Photon - "Photons are massless[...]In empty space, the photon moves at c (the speed of light) and its energy and momentum are related by E = pc, where p is the magnitude of the momentum vector p[...]Current commonly accepted physical theories imply or assume the photon to be strictly massless"

Typically momentum is thought as mass times velocity, and since photons do have momentum, there was a desire to give them some kind of "relativistic mass".

In more recent times, it has been seen as easier to use just one concept of mass, and to redefine momentum entirely. So, photons have a mass of 0, and we don't need to specify "rest mass". But they do have momentum.

Oh, thanks for the correction.