Random numbers are vital for a wide variety of applications including lotteries, statistical sampling, computer simulations and cryptography. They can also be used to make random decisions required in both quantum key distribution (QKD) and fundamental tests of the foundations of quantum mechanics. For each application the requirements on the random numbers may be different. For some applications, such as computer simulations, it is sufficient for the numbers to be statistically random without any need for them to be unpredictable. However, for many applications, especially cryptography, it is essential for the numbers to be unpredictable too. Quantum random number generators (QNRGs) offer great promise for such applications, as the unpredictability of their outputs is guaranteed by the laws of quantum mechanics, provided their implementation matches the assumptions made in their protocols. This is not always the case in practice, which can compromise the unpredictability of the QRNG's output or in other words its security. This thesis focuses on improving the security of continuous variable QRNGs. The first part of this thesis focuses on the first proof-of-principle experimental implementation of a new semi-device independent continuous variable QRNG protocol. This protocol exploits a gain-switched laser to randomly measure the quadratures of the input state, on which no assumptions are made. The main advantage of this new protocol is that it provides source device independent assurances, whilst maintaining the simple setup of a typical continuous variable QRNG. In the second part, a high bandwidth, low noise balanced homodyne detector for quantum information applications is developed. The detailed characterisation of this balanced homodyne detector leads to the identification of several implementation security flaws, which could compromise the security of a QRNG based on it or similar detectors. Finally, in the third part, one of these flaws is exploited to demonstrate the first out-of-band electromagnetic injection attack against a QRNG.