Depends on the design and size of the plane.
Small planes have no hydraulics, only mechanical links (cables, pushrods typically).
As planes get larger, hydraulics become helpful and then necessary to operate the control surfaces because the aerodynamic forces increase with larger surfaces and faster speeds.
In some cases there exists a mechanical linkage which is boosted by hydraulics, this sort of design would allow some control without hydraulics.
New planes typically have no mechanical backup on some or all controls, so hydraulics would be necessary. In those planes there are typically three hydraulic systems, 5-6 pumps, and a wind-driven backup turbine. Also some pumps are electric, but some are driven directly by the engine. In case electrical power is lost, the engine-driven pumps will still work (the fuel pumps and engines can run without any external electrical systems).
About the only thing that get through all that redundancy is actual physical damage from objects penetrating the aircraft. This is what happened with UAL flight 232, but there were some re-designs to prevent a failed engine from damaging all three hydraulic system. Today about the only that should be able to do that would be a bad collision or enemy action.
Even without hydraulics, some control might be available with differential thrust (roll/yaw), flaps/power setting (pitch), or mechanical trim tabs (if installed). That's how flight 232 arrived at the runway.