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I was always taught and lead to believe that at the beginning of our solar system, there was a period of heavy bombardment, which is conceivable.

Why has it just stopped and we are at peace from this, and, there is no such things happening today?

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According to the Nice model -- that's Nice, France (rhymes with Reese or geese), not nice (rhymes with mice) -- the solar system formed with the giant planets (the outer four) in the following order: Jupiter, Saturn, Neptune, Uranus. Outside of these planets existed a large field of asteroids (also called planetesimals), the remnants of which form our Kuiper belt.

The Nice model hypothesizes that Neptune and Uranus actually swapped places due to an orbital resonance between Jupiter and Saturn. This had the effect of moving Jupiter inward (towards the Sun) and kicking Neptune farther out. This orbital disruption also caused a thinning of the Kuiper belt, where some of the planetesimals gained kinetic energy and were knocked into the Oort cloud, and some of them slowed down and fell into the inner solar system, causing a period of heavy bombardment.

It is thought that the period of heavy bombardment ended once the orbits of the outer four planets stabilized and an equilibrium was reached. The result was the current order of the outer planets we see today (Jupiter, Saturn, Uranus, Neptune), a thinner Kuiper belt, and an outer Oort cloud. Of course, this is just a model; the Nice model itself has already undergone some revisions, even though it is relatively recent. A nice video illustrating the Nice model can be found here.

I hope this answers your question.

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