Large language model (LLM)–based agents are increasingly being deployed beyond centralized cloud environments and toward the edge of the network, where they operate closer to data sources. This transition facilitates lower latency and enhances contextual awareness, privacy, and responsiveness, but it also introduces challenges that differ from traditional cloud-based agent deployments. This survey provides a systematic overview of LLM-based edge agents with a particular focus on four critical dimensions: privacy, scalability, heterogeneity, and autonomy. To facilitate structured analysis, we introduce a novel taxonomy along four axes: deployment, functional role, interaction, and adaptation. Based on our taxonomy, we analyze the challenges LLM-based agents face on the edge and discuss design solutions that can help mitigate possible issues. We further analyze the degree to which existing LLM-based edge agent frameworks achieve privacy, scalability, heterogeneity, and autonomy.