In WDM networks using a fixed frequency grid, transmission rates can vary for different connections, leading to inefficient bandwidth utilization in optical fibers with lower-rate connections using wide frequency slots. In elastic optical networks (EONs), the frequency grid is made flexible, thereby improving the effective network capacity. A flexible frequency grid consists of smaller frequency slots, and a transmitting node can use multiple slots using suitable modulation techniques, such as optical OFDM, Nyquist-WDM and optical arbitrary waveform generation (OAWG). However, this requires bandwidth-variable transceivers (BVTs) and other devices to set up variable-rate connections. First we discuss the design challenges in EONs and describe the evolving technologies for the network elements. Then we present some offline (LP-based and heuristic) design methodologies for EONs to carry out routing and spectral allocation (RSA) for the required connections. Finally, we present some online fragmentation-aware RSA schemes for the operational EONs. (146 words)