Knowledge Base Question Answering (KBQA) involving complex reasoning is emerging as an important research direction. However, most KBQA systems struggle with generalizability, particularly on two dimensions: (a) across multiple knowledge bases, where existing KBQA approaches are typically tuned to a single knowledge base, and (b) across multiple reasoning types, where majority of datasets and systems have primarily focused on multi-hop reasoning. In this paper, we present SYGMA, a modular KBQA approach developed with goal of generalization across multiple knowledge bases and multiple reasoning types. To facilitate this, SYGMA is designed as two high level modules: 1) KB-agnostic question understanding module that remain common across KBs, and generates logic representation of the question with high level reasoning constructs that are extensible, and 2) KB-specific question mapping and answering module to address the KB-specific aspects of the answer extraction. We evaluated SYGMA on multiple datasets belonging to distinct knowledge bases (DBpedia and Wikidata) and distinct reasoning types (multi-hop and temporal). State-of-the-art or competitive performances achieved on those datasets demonstrate its generalization capability.