One issue with resistors is that you need a differential probe unless the circuit is floating or the shunt is on the low side (often inconvenient due to multiple paths to ground).
More advanced than shunts are current transformers (used by AC-only current probes). They add some inductive loading, but can have bandwidths up to a few GHz. These tend to be relatively inexpensive (of course still much more than a resistor). For DC to a few hundred kHz, Hall effect sensors are used. For DC to something like 100MHz, there are probes that combine a Hall effect sensor with a current transformer. This is extremely complex to manufacture, since any gaps in the magnetic core will screw up the linearity. I believe Tektronix and Hioki make them, and Lecroy licenses them from Hioki(?). Especially the latter ones are quite expensive.