Hypertension is known to affect around one third of adults globally and early diagnosis is essential to reduce the effects of this affliction. Todays Blood Pressure (BP) monitoring cuffs are obtrusive and inconvenient for performing regular measurements, and continuous non-invasive blood pressure devices are too complex and expensive for ambulatory use. Hence, there is a strong need for affordable systems that can measure blood pressure (BP) variations throughout the day as this will allow to monitor, diagnose and follow-up not only patients at risk, but also healthy population in general for early diagnosis. A promising method for arterial BP estimation is to measure the Pulse Transit Time (PTT) and derive pressure values from it. However, current methods for measuring this surrogate marker of BP require complex sensing and analysis circuitry and the related medical devices are expensive and inconvenient for the user. In this paper, we present new methods to estimate PTT reliably and subsequently BP, from the baseline sensors of smartphones. This new approach involves determining PTT by simultaneously measuring the time the blood leaves the heart, by recording the heart sound using the standard microphone of the phone, and the time it reaches the finger, by measuring the pulse wave using the phones camera. We present algorithms that can be executed directly on current smartphones to obtain clean and robust heart sound signals and to extract the pulse wave characteristics. We also present methods to ensure a synchronous capture of the waveforms, which is essential to obtain reliable PTT values with inexpensive sensors. Additionally, we combine Autocorrelation and Fast Fourier Transform (FFT)-based methods for reliably estimating the user heart rate (HR) from his/her heart sounds, and describe how to use the calculate HR to compensate for the camera frame rate variations and to improve the robustness of PTT estimation. Our experiments show that the computational overhead of the proposed processing methods is minimum, which allows real-time feedback to the user, and that the PTT values are fully accurate (beat-to-beat), thereby enabling state-of-the-art smartphones to be used as affordable medical devices.