Large Language Models (LLMs) are revolutionizing software engineering (SE), with special emphasis on code generation and analysis. However, their applications to broader SE practices including conceptualization, design, and other non-code tasks, remain partially underexplored. This research aims to augment the generality and performance of LLMs for SE by (1) advancing the understanding of how LLMs with different characteristics perform on various non-code tasks, (2) evaluating them as sources of foundational knowledge in SE, and (3) effectively detecting hallucinations on SE statements. The expected contributions include a variety of LLMs trained and evaluated on domain-specific datasets, new benchmarks on foundational knowledge in SE, and methods to detect hallucinations. Initial results in terms of performance improvements on various non-code tasks are promising.