MmWave communication is a promising technology for 5G wireless systems. However, the large number of antennas used and the wide signal bandwidth in mmWave systems render the conventional multi-antenna techniques increasingly costly in terms of signal processing complexity, hardware implementation, and power consumption. In this article, we investigate cost-effective mmWave communications by first providing an overview of the main existing techniques that offer different trade-offs between performance and cost, and then focusing our discussion on a promising new technique based on the advanced lens antenna array. It is revealed that by exploiting the angle-dependent energy focusing property of lens arrays, together with the angular sparsity of the mmWave channels, mmWave lens antenna system is able to achieve capacity-optimal performance with very few RF chains and using low-complexity single-carrier transmission, even for wideband frequency-selective channels. Numerical results show that the lens-based system outperforms the benchmark schemes for mmWave communications in both spectrum efficiency and energy efficiency.