Yes. They can be made through either fusion or fission.
Through fusion, atoms with lower molecular numbers can combine to make one atom of a higher molecular number. This process occurs in stars when hydrogen is fused into helium.
In nuclear fission, an atom with a high molecular number, looses neutrons and protons to become an atom with a lower molecular number. It is said to be radioactive, and this happens naturally as uranium turns into lead. When it happens naturally the rate at which it goes through fission is called a half-life. When it is used in a nuclear reactor it can release energy as if undergoes fusion and produce a lot of energy over a long period of time.
When it is used in an atomic bomb, the fission undergoes a chain reaction where all the energy is released in a fraction of a second and creates a huge explosion.
When it is used in a nuclear bomb, an atom bomb explodes under nuclear fission and then triggers hydrogen to fuse into helium in a secondary explosion. This type of bomb releases even more energy.
So. Yes, atoms can be made and destroyed, in both cases huge amounts of energy are released in the process.