The surface structures and growth kinetics of InGaN(0001) are studied. It is well known that during molecular beam epitaxy GaN surfaces undergo a smooth to rough transition when the growth condition is switched from Ga rich to N rich. It is found here that indium atoms have only a small effect on this transition when deposited on GaN(000 (1) over bar), but when deposited on GaN(0001) the indium acts as a surfactant and greatly extends the regime of smooth growth. Near the smooth/rough transition of InGaN(0001) growth, a bright root 3x root 3 reconstruction is observed at growth temperature. The formation kinetics of this reconstruction are studied in detail. Scanning tunneling microscopy and total energy computations are used to study the structure of InGaN(0001) surfaces under metal rich conditions. Indium is found to occupy the top two atomic layers of the crystal; its incorporation in the second layer produces significant strain, leading to the formation of small pits on the surface and increased indium concentration inside and around the pits.