In today's data-driven landscape, time series forecasting is pivotal in decision-making across various sectors. Yet, the proliferation of more diverse time series data, coupled with the expanding landscape of available forecasting methods, poses significant challenges for forecasters. To meet the growing demand for efficient forecasting, we introduce auto-sktime, a novel framework for automated time series forecasting. The proposed framework uses the power of automated machine learning (AutoML) techniques to automate the creation of the entire forecasting pipeline. The framework employs Bayesian optimization, to automatically construct pipelines from statistical, machine learning (ML) and deep neural network (DNN) models. Furthermore, we propose three essential improvements to adapt AutoML to time series data: First, pipeline templates to account for the different supported forecasting models. Second, a novel warm-starting technique to start the optimization from prior optimization runs. Third, we adapt multi-fidelity optimizations to make them applicable to a search space containing statistical, ML and DNN models. Experimental results on 64 diverse real-world time series datasets demonstrate the effectiveness and efficiency of the framework, outperforming traditional methods while requiring minimal human involvement.