This paper proposes cuffless and continuous blood pressure estimation utilising Photoplethysmography (PPG) signals and state of the art recurrent network models, namely, Long Short Term Memory and Gated Recurrent Units. The models were validated on wide range of varying blood pressure and PPG signals acquired from the Multiparameter Intelligent Monitoring in Intensive Care database. Many features were extracted from the PPG waveform and several machine learning techniques were employed in an attempt to eliminate collinearity and reduce the size of input feature vector. Consequently, the most effective features for blood pressure estimation were selected. Experimental results show that the accuracy of the proposed methods outperform traditional models applied in the literature. The results satisfy the American National Standards of the Association for the Advancement of Medical Instrumentation.