Although concrete is more eye appealing then other paving surfaces, it does require repair. Concrete is constantly absorbing water and does break down over time or due to other problems such as drainage. The following will help you in your efforts to repair a concrete sidewalk.
Clean the crack or hole in the cement sidewalk. Use a hose to thoroughly spray in the hole to remove dirt, pebbles, chunks of broken cement and such. Scrub the crack or hole with a stiff brush.
Enlarge the hole into a V shape by chiseling it with a cold chisel and hammer.
Clean again making sure all debris is removed from the hole.
Fill the bottom of the hole with sand if the hole is deeper than half an inch.
Mix cement and water into a thick paste.
Dampen the crack, the hole (even if it has sand in the bottom) and the surrounding cement area with a hose.
Fill the area with the cement mixture. Use a caulking gun for cracks and a putty knife to get the cement into the holes.
Fill it above the surface so it can settle. Lightly tap the top surface.
Scrape away the excess cement mixture with the putty knife so it is level with its surroundings.
Leave it alone for 2 hours.
Spray the area and its surroundings again.
Cover with plastic for 3 days. Spray it each day until the cover is removed.
Repeat the cement filler if the new cement settles below the surface.