Text simplification (TS) refers to the process of reducing the complexity of a text while retaining its original meaning and key information. Existing work only shows that large language models (LLMs) have outperformed supervised non-LLM-based methods on sentence simplification. This study offers the first comprehensive analysis of LLM performance across four TS tasks: lexical, syntactic, sentence, and document simplification. We compare lightweight, closed-source and open-source LLMs against traditional non-LLM methods using automatic metrics and human evaluations. Our experiments reveal that LLMs not only outperform non-LLM approaches in all four tasks but also often generate outputs that exceed the quality of existing human-annotated references. Finally, we present some future directions of TS in the era of LLMs.