What occurs when waves overlap??

Interference, the two main types of interference are constructive (where the waves meet together, i.e. the peak with the peak) or destructive (where the waves meet and destroy each other, i.e. the peak with the trough).

You can think of interference as a new wave being created with the amplitude of it being the sum of the two amplitudes that the other waves would have had at that point.

Depends on the nature of the overlapping waves. They might enhance each other...constructive interference. They might negate each other...destructive interference. Or if they are totally different frequencies they might simply pass through each other with little to no effect.

There is a principle of SUPERPOSITION.
At any point where two or more waves meet the amplitude of the wave at that point and at that time is the sum of the amplitudes of the individual waves.

Now the wave can change over time as in the doppler effect.
or it can change over position as in the two slit interference.

Yet at every position and every instant the rule above applies.