This paper presents Neuro-Evolutionary Optimization SLAM (NeoSLAM) a novel approach to SLAM that uses a neural network (NN) to autonomously learn both a nonlinear motion model and the noise statistics of measurement data. The NN is trained using evolutionary optimization to learn the residual error of the motion model, which is then added to the odometry data to obtain the full motion model estimate. Stochastic optimization is used, to accommodate any kind of cost function. Prediction and correction are performed simultaneously within our neural framework, which implicitly integrates the motion and sensor models. An evolutionary programming (EP) algorithm is used to progressively refine the neural model until it generates a trajectory that is most consistent with the actual sensor measurements. During this learning process, NeoSLAM does not require any prior knowledge of motion or sensor models and shows consistently good performance regardless of the robot and the sensor noise type. Furthermore, NeoSLAM does not require the data association step at loop closing which is crucial in most other SLAM algorithms, but can still generate an accurate map. Experiments in various complex environments with widely-varying types of noise show that the learning capability of NeoSLAM ensures performance that is consistently less sensitive to noise and more accurate than that of other SLAM methods.