Recent experimental results and proposals towards implementation of quantum teleportation are discussed. It is proved that reliable (theoretically, 100% probability of success) teleportation cannot be achieved using the methods applied in recent experiments, i.e., without quantum systems interacting with one another. Teleportation proposals involving atoms and electromagnetic cavities are reviewed and the most feasible methods are described. In particular, the language of nonlocal measurements has been applied; this language has also been used for presenting a method for teleportation of quantum states of systems with continuous variables.