Electricity load and price data pose formidable challenges for forecasting due to their intricate characteristics, marked by high volatility and non-linearity. Machine learning (ML) and deep learning (DL) models have emerged as promising tools for effectively predicting data exhibiting high volatility, frequent fluctuations, mean-reversion tendencies, and non-stationary behavior. Therefore, this review article is dedicated to providing a comprehensive exploration of the application of machine learning and deep learning techniques in the context of electricity load and price prediction. In contrast to existing literature, our study distinguishes itself in several key ways. We systematically examine ML and DL approaches employed for the prediction of electricity load and price, offering a meticulous analysis of their methodologies and performance. Furthermore, we furnish readers with a detailed compendium of the datasets utilized by these forecasting methods, elucidating the sources and specific characteristics underpinning these datasets. Then, we rigorously conduct a performance comparison across various performance metrics, facilitating a comprehensive assessment of the efficacy of different predictive models. Notably, this comparison is carried out using the same datasets that underlie the diverse methodologies reviewed within this study, ensuring a fair and consistent evaluation. Moreover, we provide an in-depth examination of the diverse performance measures and statistical tools employed in the studies considered, providing valuable insights into the analytical frameworks used to gauge forecasting accuracy and model robustness. Lastly, we devote significant attention to the identification and analysis of prevailing challenges within the realm of electricity load and price prediction. Additionally, we delve into prospective directions for future research, thereby contributing to the advancement of this critical field.