In 1805 at the end of November, Napoleon decided to launch one more plan to lure the Austrians and Russians into battle, by appearing to order his forces to retreat and letting it be known that he did not want to face his enemies on the field. Eventually, the Allies fell for his plan and decided to attack what they saw as a weakening force, just outside the town of Austerlitz.
Napoleon was weaker in numbers, but had had time to plan and examine the battlefield closely; he had his enemies where he wanted them.
It was a fiercely contested battle, but Napoleon had the upper hand and destroyed a third of the Russian-Austrian force, with 15,000 casualties and 12,000 prisoners, in comparison he had lost c.8,000 men. He was victorious and regarded it as one of his greatest triumphs, paving the way for French dominance in Europe for the next decade.
To understand what happened and why read Battle Story.