We study the packing of a large number of congruent and non-overlapping circles inside a regular polygon. We have devised efficient algorithms that allow one to generate configurations of N densely packed circles inside a regular polygon, and we have carried out intensive numerical experiments spanning several polygons (the largest number of sides considered here being 16) and up to 200 circles (400 circles in the special cases of the equilateral triangle and the regular hexagon). Some of the configurations that we have found possibly are not global maxima of the packing fraction, particularly for N≫1, due to the great computational complexity of the problem, but nonetheless they should provide good lower bounds for the packing fraction at a given N. This is the first systematic numerical study of packing in regular polygons, which previously had only been carried out for the equilateral triangle, the square, and the circle.