Alzheimer's disease (AD) is considered as one of the leading causes of death among people over the age of 70 that is characterized by memory degradation and language impairment. Due to language dysfunction observed in individuals with AD patients, the speech-based methods offer non-invasive, convenient, and cost-effective solutions for the automatic detection of AD. This paper systematically reviews the technologies to detect the onset of AD from spontaneous speech, including data collection, feature extraction and classification. First the paper formulates the task of automatic detection of AD and describes the process of data collection. Then, feature extractors from speech data and transcripts are reviewed, which mainly contains acoustic features from speech and linguistic features from text. Especially, general handcrafted features and deep embedding features are organized from different modalities. Additionally, this paper summarizes optimization strategies for AD detection systems. Finally, the paper addresses challenges related to data size, model explainability, reliability and multimodality fusion, and discusses potential research directions based on these challenges.