Just to add a few elaborations to John T's excellent replies:
As the wing approaches M1 the local mach no over the top of the wing will reach M1 before the rest of the aircraft does (faster local airflow over the wing). The shockwave normally starts to form at about the half chord position and then moves aft as speed increases. At about the same time the leading edge shock also forms. Eventually the shock that started to form at the half chord position moves right back to the trailing edge of the wing and once the aircraft is fully supersonic, there will be a very clear shock wave attached to the wing leading edge, and another attached to the trailing edge.
The problems come during the transonic phase, as the aircraft transitions from subsonic to supersonic speed. For example, mach tuck (as already mentioned) due to the pressure distribution on the wing moving aft. Not only does this cause a nose-down pitching moment, but it also changes the directional stability and pitch stability. Also, as the wing shockwave moves aft it can cause control problems such as control reversal, excessively large control forces, or control surface failure.
If you look at modern supersonic aircraft, most have all-moving tailplanes (or canards and/or elevons if they have a delta wing) driven by powerful hydraulic (irreversible) flight control actuators to ensure control is retained throughout the transonic phase.