In gymnastics, a cartwheel is a sideways rotary movement performed by bringing the hands to the ground while the body inverts and the legs travel over the body, coming down to a standing position.
The name cartwheel is used because, when a human performs this, their limbs move like the spokes of a turning wheel. In classical Indian dance it is called Talavilasitam karana. The martial art of capoeira has a similar move called the aú.
A cartwheel is carried out when one moves sideways (in the motion the wheel of a cart would follow) in a straight line keeping the back straight placing the hand of the same side on the ground followed by the other hand as the legs are passed over the body and then come down as the hands and body come up to standing. The cartwheel can also be performed using one or no hands. A cartwheel using no hands is called an aerial and requires much more momentum and speed.
A classic cartwheel performed with proper gymnastics form always starts with a lunge, lead leg (stronger leg) in the front of the lunge and the weaker leg in the back. During the lunge the gymnast will have their arms high in the air and straight, and hips are square facing the forward direction. The gymnast will then push off of their front leg of the lunge, followed by placing their hands side by side on the ground in front of them. As they do this they will begin to kick their legs up and over their torso and head as the body becomes inverted. During the rotation the legs stay apart in a large, wide straddle(as far apart as the gymnast can get them), legs are straight, and toes and feet always stay pointed. During the rotation legs must reach at least 90 degrees of motion in a competition in order to get points (USA). Finally the gymnast will set their first foot on the ground, followed by the second foot, landing in a lunge with the weaker leg in the front of the lunge and the lead leg will be in back. The gymnast will land facing the opposite direction than they started in. Their hands and arms will be perfectly straight, pointed high in the sky.
Cartwheels have a long tradition in the city of Düsseldorf, Germany. According to the legend, upon hearing that their city had won the Battle of Worringen in 1288, the children of Düsseldorf started performing flips in celebration. Sooner or later, they began performing side flips. Then the flips became cartwheels.
Cartwheel progressions 
Chasse cartwheel: the gymnast will perform some side step gallops before entering into the cartwheel.
Cartwheel, 1/2 turn, Cartwheel: the gymnast will perform a cartwheel, then after landing in the proper lunge position they will turn their body to face the direction that they are headed, while staying in their lunge. Then they will perform another cartwheel
One handed cartwheel (close handed): The gymnast will perform a cartwheel, but will only place the first hand on the ground. The following arm will be placed by the gymnast's side and behind their back.
One handed cartwheel (far handed): The gymnast will perform a cartwheel, but only place the second hand on the ground. The first one will be placed by the gymnast's side and behind their back.
Aerial (no handed cartwheel): The gymnast will start in a lunge with their lead leg in the back. They will then bring their lead leg up into the air with power as if they were doing one skip, this is called a hurdle. The gymnast will then drive the lead leg down as the lead foot pushes off the ground as it lands. During that motion the gymnast will kick their back following leg up, as they continue into a cartwheel motion. Except rather than placing the hands on the ground the gymnast will swing their arms up, and then shoot them down to their sides. The kick and push of the legs combined with the rotation and punch to the sides of the arms will force the gymnasts body up into the air in a side rotation. Then the gymnast will land the aerial the same way as a cartwheel.
Cartwheels and beam 
The cartwheel is a skill commonly performed on the balance beam apparatus. It is a required skill in the new USAG Level 4 Compulsory beam routine. Aerial cartwheels are also frequently performed among gymnasts levels 9 and 10.