We design polar codes of a growing length n and a code rate R →1 that have polynomial complexity of order nlog n in code construction, encoding and decoding. These codes 
achieve the vanishing output error rates on the binary symmetric channels with the transition error probabilities p→0 and require  a substantially smaller redundancy (1-R)n than do other known high-rate codes,such as Reed-Muller or BCH codes. We then consider the low-rate polar codes that achieve the vanishing output error rates with an asymptotically optimal decline in code rate R→0.