This chapter deals with some of the most prominent successes of quantum computing. The most well-known quantum computing algorithm, Shors algorithm for factoring a number in its prime factors, is discussed in details. The key to Shors algorithm is the quantum Fourier transform that is explained with the help of simple examples. The role of quantum entanglement is also discussed. The next important quantum computing algorithm is Grovers algorithm that helps in searching an item in an unsorted database. This algorithm is motivated by first discussing a quantum shell game in which a pea hidden under one of the four shells is found in one measurement with certainty each time. This amazing result is then generalized to an arbitrary number of objects and Grovers algorithm.