Fine-tuning pretrained LLMs has proven effective for reaching state-of-the-art performance on specific tasks like machine translation. However, this process often implies sacrificing general-purpose capabilities, such as conversational reasoning and instruction-following, hampering the usefulness of the system in real-world applications requiring a mixture of skills. In this paper, we introduce Tower+, a suite of models designed to deliver strong performance on both translation and multilingual general-purpose text capabilities. We improve the TOwer (Alves et al., 2024) recipe by adding novel stages of preference optimization and reinforcement learning with verifiable rewards, in addition to continued pretraining and supervised fine-tuning. At each stage, we carefully generate and curate data to strengthen performance on translation and general-purpose tasks like coding, mathematics, and instruction-following. We develop models at multiple scales: 2B, 9B, and 72B. Our smaller models often outperform larger general-purpose open-weight and proprietary LLMs (e.g., Llama 3.3 70B, GPT-4o). Our largest model delivers best-in-class translation performance for high-resource languages, and top results on multilingual Arena Hard and IF-MT, a benchmark we introduce for evaluating both translation and instruction-following.