Thanks everybody for useful information.
Now, if I understand correctly, the problem can be diagnosed as "the house is put too low". During heavy rains the water rises above the basement level, and finds its way through the concrete into the basement :(
As a side note, in the last 5 years that I own the house, there was one more issue in the basement -- a small pool of water in the middle of the basement, apparently coming through the floor.
Also, the air in the basement is normally relatively wet -- when I run a dehumidifier, it gathers a couple gallons of water a day, but still can't get the humidity below 70%. Can this be another sign of close water?
So, some measures, like correcting downspouts, making sure the gutters are clean, and there are no slopes toward the house can aleviate the problem, but can't solve it -- when the water table is above the basement level (due to heavy rain, etc.), the water will find its way into the basement...
Where it needs to be pumped out with a sump pump. This is basically the same thing what I am doing right now with my pool pump, but can be done better (and more conveniently) since the sump pump is built in, located below floor level, operates automatically, etc.
Is this correct?
Also, what about this waterproof paint that is advertised at some sites? Is it any effective at all?
Thanks, Arkadiy