As artificial intelligence (AI) systems advance, we move towards broad AI: systems capable of performing well on diverse tasks, understanding context, and adapting rapidly to new scenarios. A central challenge for broad AI systems is to generalize over tasks in related domains and being robust to distribution shifts. Neuro-symbolic (NeSy) AI bridges the gap between symbolic and sub-symbolic paradigms to address these challenges, enabling adaptable, generalizable, and more interpretable systems. The development of broad AI requires advancements in domain adaptation (DA), enabling models trained on source domains to effectively generalize to unseen target domains. Traditional approaches often rely on parameter optimization and fine-tuning, which can be impractical due to high costs and risks of catastrophic forgetting. NeSy AI systems use multiple models and methods to generalize to unseen domains and maintain performance across varying conditions. We analyze common DA and NeSy approaches with a focus on deep domain-invariant learning, extending to real-world challenges such as adapting to continuously changing domains and handling large domain gaps. We showcase state-of-the-art model-selection methods for scenarios with limited samples and introduce domain-specific adaptations without gradient-based updates for cases where model tuning is infeasible. This work establishes a framework for scalable and generalizable broad AI systems applicable across various problem settings, demonstrating how symbolic reasoning and large language models can build universal computational graphs that generalize across domains and problems, contributing to more adaptable AI approaches for real-world applications.