In the context of optical signal processing, quantum and quantum-inspired machine learning algorithms have massive potential for deployment. One of the applications is in error correction protocols for the received noisy signals. In some scenarios, non-linear and unknown errors can lead to noise that bypasses linear error correction protocols that optical receivers generally implement. In those cases, machine learning techniques are used to recover the transmitted signal from the received signal through various estimation procedures. Since quantum machine learning algorithms promise advantage over classical algorithms, we expect that optical signal processing can benefit from these advantages. In this review, we survey several proposed quantum and quantum-inspired machine learning algorithms and their applicability with current technology to optical signal processing.