Optimal order placement is a key aspect of market making, and more generally, of liquidity providing strategies in electronic markets. With this motivation in mind, we study the optimal placement of limit orders from theoretical and numerical points of view, in the context of Markovian limit order book models. The theoretically optimal strategies are then backtested using real data, providing results that advocate for the design of better order book models. Some extensions are made, based either on Hawkes processes, or on processes with finite memory (joint works with C. Huré, X. Lu, H. Pham).