We present, BSense, a flexible broadband mapping system for broadband coverage and quality assessment of broadband connections in a given geographic region. For coverage related analysis, it relies on data that is either obtained from ISPs or generated based on technology models and information about infrastructure sites. Broadband quality assessment in BSense is via host-based measurements using our multi-platform and technology-adaptive software client which periodically runs as a background process on users' computers. The host-based software measurement approach employed in BSense is not only cost-effective but is also flexible and reduces measurement bias. BSense also incorporates a flexible broadband quality index for summarizing the collective effect of various underlying attributes such as download/upload speeds and latency. BSense system has been implemented based on open-source software components. We conduct extensive evaluations of the measurement component of BSense aimed at quantifying system effects and to compare against other measurement techniques and approaches, the latter showing that BSense is fairly close to sophisticated and expensive alternatives. The usefulness of the BSense system is demonstrated using two real world case studies, one on identifying notspots in Scotland and the other on broadband quality assessment in a rural part of Scotland through pilot deployment.