On the one hand, processors for hearing aids are highly specialized for audio processing, on the other hand they have to meet challenging hardware restrictions. This paper aims to provide an overview of the requirements, architectures, and implementations of these processors. Special attention is given to the increasingly common application-specific instruction-set processors (ASIPs). The main focus of this paper lies on hardware-related aspects such as the processor architecture, the interfaces, the application specific integrated circuit (ASIC) technology, and the operating conditions. The different hearing aid implementations are compared in terms of power consumption, silicon area, and computing performance for the algorithms used. Challenges for the design of future hearing aid processors are discussed based on current trends and developments.