Electricity load forecasting is a key task in the planning and operation of power systems and electricity markets, and its importance increases with the advent of smart grids. In this paper, we present AWNN, a new approach for very short-term load forecasting. AWNN decomposes the complex electricity load data into components with different frequencies that are predicted separately. It uses an advanced wavelet transform with entropy cost function to select the best wavelet basis for data decomposition, mutual information for feature selection and neural networks for prediction. The performance of AWNN is comprehensively evaluated using Australian and Spanish electricity load data for one-step and multi-step ahead predictions, and compared with a number of benchmark algorithms and baselines.