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A covalent bond is formed when the electron clouds of two atoms overlap. Where the clouds overlap they are thicker, and their electric charge is stronger.
Both nuclei feel a force of attraction towards this thick electron cloud, and so the two atoms are held together. A group of atoms joined in this way is known as a molecule.
The number of covalent bonds which an atom can form depends on how many electrons it has. The shell model explains this.
 

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