Deep learning (DL) methods have brought world-shattering breakthroughs, especially in computer vision and classification problems. Yet, the design and deployment of DL methods in time series prediction and nonlinear system identification applications still need more progress. In this paper, we present DL frameworks that are developed to provide novel approaches as solutions to the aforementioned engineering problems. The proposed DL frameworks leverage the advantages of autoencoders and long-short term memory network, which are known being data compression and recurrent structures, respectively, to design Deep Neural Networks (DNN) for modeling time series and nonlinear systems with high performance. We provide recommendations on how deep AEs and LSTMs should be utilized to end up with efficient Prediction-focused (Pf) and Simulation-focused (Sf) DNNs for time series and system identification problems. We present systematic learning methods for the DL frameworks that allow straightforward learning of Pf-DNN and Sf-DNN models in detail. To demonstrate the efficiency of the developed DNNs, we present various comparative results conducted on the benchmark and real-world datasets in comparison with their conventional, shallow, and deep neural network counterparts. The results clearly show that the deployment of the proposed DL frameworks results with DNNs that have high accuracy, even with a low dimensional feature vector.