Aligned large language models (LLMs) are vulnerable to jailbreaks, which bypass the safeguards of targeted LLMs and fool them into generating objectionable content. While initial defenses show promise against token-based attacks, there are no defenses that provide robustness against semantic attacks and avoid unfavorable trade-offs between robustness and nominal performance. To meet this need, we propose SemanticSmooth, a smoothing-based defense that aggregates the predictions of multiple semantically transformed copies of a given input prompt. Experimental results demonstrate that SemanticSmooth achieves strong robustness against both manually constructed jailbreak prompts and automatic jailbreak attacks like GCG, PAIR, and PromptRS while maintaining strong nominal performance on standard LLM evaluation benchmarks such as AlpacaEval for the instruction-following tasks and PiQA for the question-answering tasks.