Large Language Models (LLMs) have recently garnered significant attention, primarily for their capabilities in text-based interactions. However, natural human interaction often relies on speech, necessitating a shift towards voice-based models. A straightforward approach to achieve this involves a pipeline of ``Automatic Speech Recognition (ASR) + LLM + Text-to-Speech (TTS)", where input speech is transcribed to text, processed by an LLM, and then converted back to speech. Despite being straightforward, this method suffers from inherent limitations, such as information loss during modality conversion and error accumulation across the three stages. To address these issues, Speech Language Models (SpeechLMs) -- end-to-end models that generate speech without converting from text -- have emerged as a promising alternative. This survey paper provides the first comprehensive overview of recent methodologies for constructing SpeechLMs, detailing the key components of their architecture and the various training recipes integral to their development. Additionally, we systematically survey the various capabilities of SpeechLMs, categorize the evaluation metrics for SpeechLMs, and discuss the challenges and future research directions in this rapidly evolving field.