Kip Up (Monkey flip): While this is not strictly a Capoeira move, it is used by many players of the Contemporary style. To learn this move start by practising a backwards roll (never do this in the roda, though!) to get a feel for the correct starting position. Next, do a kind of half backwards roll, just getting onto your shoulders, and with your hands flat on the floor and your legs above your head. The rest of your back should not be touching the floor. Do this a few times to get used to it. When you are comfortable, try the actual kip-up. You have to arch your back strongly while pushing off with your hands. Try to kick your legs more up than forward so that you land as upright as possible.
Because the back never touches the floor in Capoeira, use the following to get into the position to kip up:
Queda de Rins to kip up: In queda de rins position, your weight is mostly on one arm. Push with the other arm so that you roll onto your shoulders. Make sure you are high on your shoulders, and that only your shoulders, and not the rest of your back.
Neckspring: Before you try this, do a couple of forward rolls. Next, do a "half" forward roll, just getting onto your shoulders. This is the position to kip up. The most important thing with this movement is to do it slowly.
Macaco to kip up: To do this, its good if you can control your macaco so that you can do it into a handstand. When doing this movement, macaco back, then when both hands are on the ground, slowly bend your arms and tuck your chin into your chest until you are in postion to kip up. Practise this on a soft surface!
Flic flac to kip up: You must have a very good and confident flic flac to do this move. Practise flipping back into a handstand, once you can do this, do the same as in the Macaco to kip up: once both hands are on the ground, lower your body slowly into the kip up starting position with your head tucked into your chest. I recommend putting a crashmat behind you when you first learn, so that you don't break you neck when you are first learning.

Relogio: Drop into an esquiva, then Rolę over your straight leg. The other leg comes over so that you are in a sort of push-up position, but with your legs wide apart while you place the hand which wasn't already on the ground. Continue the rolling motion by lowering your torso onto the elbow of that arm, so that it supports all your weight (like in a Queda de Rins). Keep rolling a bit more so that your body is side on to the ground. You spin around by swinging the leg that is furthest from the ground, back. Try to balance on the support arm and push off with the other one. You should move around like the hands of a clock (relogio in Portuguese), then get back onto your feet as you finish going around. The more you spin, the better it looks.

Aú Giratoria: You begin the movement as if you are doing a Meia Lua de Compasso (called Rabo de Arraia in some schools), that is you turn around with your weight on one foot, the right foot if you are doing the movement anti-clockwise, or the left if you are doing the movement clockwise. You then place the same hand (the right hand if you are doing the movement anti-clockwise or vice versa) on the ground, tranferring your weight from your foot onto that hand. While doing this you swing the leg which has no weight on it around and up behind you. That movement along with the weight transfer and also by pushing off with the grounded leg should get you into a sort of one hand handstand position. To balance, your other hand should now come down onto the ground. Try to let your body spin on the first hand a little before placing the next one. You can keep walking yourself around as much as you want by continuing to transfer your weight from hand to hand, usually in 45 degree steps. This will be easy if you can walk well forward and back on your hands.

S-Dobrado (Chapéu de Couro, Doublé-S): Drop into an esquiva, with the supporting arm close to the knee of your front leg. Sweep the back leg around while shifting your weight onto the other leg, and swapping supporting hands. Once the sweeping leg gets in the front of you, get onto your hands by arching back in a Macaco motion, flicking your head straight back. The rest of the motion is the same as a Macaco. Once you have mastered the movement, try it slow and with complete control.

S-Dobrado (Chapéu de Couro, Doublé-S) to Aú Giratoria: Drop into an esquiva, with the supporting arm close to the knee of your front leg. Sweep the back leg around while shifting your weight onto the other leg, and swapping supporting hands. Once the sweeping leg gets to the front get onto your hands by arching back in a Macaco type motion, but with the head going slightly towards the grounded hand instead of straight back like in a Macaco. To keep turning, just move like in the Aú Giratoria (above).
Variations:

• Handstand spin aka 1990 aka Piăo de Măo, keep your legs apart as you raise into the handstand, then bring them together in a circular motion after you have transferred your weight onto your second hand, which you should then be spinning on. The best place to spin on your hand is on your palm heel, opposite your thumb.
• Headspin aka Piăo de Cabeça, keep your legs apart as you raise into the handstand, lower onto your head then bring them together in a circular motion. Only try this if you already know how to headspin.
• Elbow spin, just like the handstand spin, but lower onto your elbow by bending your arm. Its hard to spin because to do that, you have to keep your forearm off the ground.
• Finish in Aú Batido.

Macaco Aú (Macaco em pé): Lower back onto one hand, kicking up the opposite leg. Unless you've got awesome flexibility and strength you will probably need to jump into this. Follow the kicking leg by pushing off your support leg. The timing is important. Follow the Macaco motion through to land one leg at a time, the second leg stepping forward into the first position (you may end up in front of the place where you started). This movement can be done continuously.

Handspring (Rabo de Arraia): Before you do this, you have to have a good handstand. To get the initial feel of it, do a handstand and keep going over onto a mat. If you can, try a handstand into a bridge. When doing the handspring, take a small hop on the back leg, then in a really fast motion, step onto the front leg while kicking the other leg back (It helps give you height, so throw it up as high as it can go) and placing the hands. The closer to your feet your hands are placed, the easier. From here keep your back leg going overhead while pushing of the support leg like you're doing a fast handstand, bringing both legs together above you. This is the hard part. Once you're at the third step in the picture, you arch your back, with your legs still going forward while pushing off with your shoulders. It will take alot of practice to get this part down. Try not to use your elbows as its much less springy that way. At first though, just practise going over and trying to land on your feet. NOTE: I haven't put how to do the back handspring because its hard to learn on your own.
Variations:

• Walkout: Bring your legs together only slightly in the air, and keep going forward, trying to land on your kicking leg. This is one of my favourites because it can be done slowly and controlled, and combined with other moves. For the Capoeira kick version, the Compasso, use only one hand, and bend the kicking leg when its in the air. The pushing leg should come down as an overhead kick.
• Aerial walkover: Handspring walkout with no hands. As above, but without the help of your arms to get you over. I can do this sort of, but I land in a crouch, like the way Paul Phoenix does in Tekken. To do it properly, you need to be very supple and flexible in your hips.
• Double leg takeoff (Flyspring/Punchfront): This one's hard. Jump onto your hands off both feet. It just shows how much easier the kick back makes this move.

Meia Lua Reversăo: This move is good for beating a rasteira. The beginning is just like a Rabo de Arraia/Meia Lua de Compasso, but you use the momentum of the kick plus a hard push up from your support leg to get you off your feet. You then twist your hips in the direction of the kick while pushing off with your shoulders. Keep your legs spread and try to land on your kicking leg. Keep going with the momentum so that you end up facing your original direction. The move is easier if you know how to do a front handspring because the kick back, and push-off timing are basically the same. Once you can do it, try to keep your eyes on your opponent as you do it.

Helicoptero: This is a stylish variation of the Aú, the difference being that you land on the opposite leg. For example if you are going left, you land on your left leg, if you go right, you land on the right. The motion of the legs is similar to a Piăo de Măo (handspin), but you should just think about moving your landing leg back, the rest will be done automatically. Really push hard off your hands and takeoff foot, because to lessen the impact on your landing leg, which can be really painful. When you land, you follow your momentum around like in the Meia Lua Reversăo. Really push hard off your hands and takeoff foot, because to lessen the impact on your landing leg, which can be really painful. Once you have it down, though, try to make the leg movements wide, circular and fast.