We present a comprehensive, in-depth review of ideation assisted by large language models (LLMs), highlighting emerging trends and identifying unaddressed research gaps. In total, we examined 61 studies investigating the application of LLMs in both group and individual ideation processes. From these studies, we derived the Hourglass Ideation Framework for LLM-assisted ideation, comprising three phases and seven key ideation stages, which served as the basis for our systematic survey. Our analysis reveals that LLMs are most frequently used for idea generation and refinement, but their use in scope specification, foundational material structuring and multi-idea evaluation and selection remains limited. We provide our findings in extensive tabular and online formats. These catalogues detail research on LLM-assisted, purely LLM-based, and human-only activities across the seven ideation stages for each of the 61 studies. These also detail creative domains, publication outlets, interaction designs, user study designs, and assessment methods. Our analysis of system interaction design reveals a predominant focus on supporting individual ideation activities and text-based interaction, with a growing trend of incorporating multimedia elements. However, in group ideation, tools and interaction modalities targeting both synchronous and asynchronous collaboration are much scarcer. We synthesize the primary findings of our review and outline promising directions for future research in LLM-assisted ideation. We hope this review will help researchers quickly gain an overview of this rapidly expanding area, efficiently locate relevant work, and identify underexplored areas for further investigation. In addition, we believe the framework we present here will form the basis for the development of future problem and solution space taxonomies, and methodologies for LLM-assisted ideation development and use.