Modern signal processing (SP) pipelines, whether model-based or data-driven, often constrained by complex and fragmented workflow, rely heavily on expert knowledge and manual engineering, and struggle with adaptability and generalization under limited data. In contrast, Large Language Models (LLMs) offer strong reasoning capabilities, broad general-purpose knowledge, in-context learning, and cross-modal transfer abilities, positioning them as powerful tools for automating and generalizing SP workflows. Motivated by these potentials, we introduce SignalLLM, the first general-purpose LLM-based agent framework for general SP tasks. Unlike prior LLM-based SP approaches that are limited to narrow applications or tricky prompting, SignalLLM introduces a principled, modular architecture. It decomposes high-level SP goals into structured subtasks via in-context learning and domain-specific retrieval, followed by hierarchical planning through adaptive retrieval-augmented generation (RAG) and refinement; these subtasks are then executed through prompt-based reasoning, cross-modal reasoning, code synthesis, model invocation, or data-driven LLM-assisted modeling. Its generalizable design enables the flexible selection of problem solving strategies across different signal modalities, task types, and data conditions. We demonstrate the versatility and effectiveness of SignalLLM through five representative tasks in communication and sensing, such as radar target detection, human activity recognition, and text compression. Experimental results show superior performance over traditional and existing LLM-based methods, particularly in few-shot and zero-shot settings.