Tephra hazards can cause widespread impacts to buildings which reduce available housing and commercial activities during post-eruption recovery and create a large burden on insurance and construction sectors. Impacts can be reduced by undertaking loss assessments to inform pre-eruption recovery planning but limited understanding of how vulnerable buildings are to certain aspects of tephra hazard makes accurate loss assessments challenging. The objectives of this thesis are to develop quantitative methods to assess building vulnerability to ballistic and tephra fall hazards which can then be used to inform tephra impact assessments in Auckland, New Zealand. To achieve these objectives building vulnerability has been quantified through the development of fragility and vulnerability functions. These functions can be used to relate hazard intensity (i.e. tephra fall thickness) with what assets are exposed (i.e. residential buildings) to predict impact. Development of new fragility functions has used data from post-eruption building surveys at Mt Ontake and Mt Usu, review of relevant literature, and novel ballistic cannon experiments. A total of 93 experiments were carried out on 4 roof and wall claddings commonly used in New Zealand construction, impacted by dense volcanic rocks moving at up to 39 m/s. The claddings were highly vulnerable to perforation by perpendicular impacts but were able to deflect oblique impacts to receive significantly less damage. The velocity of building material fragments ejected during experiments was also measured with implications for life safety. Conducting and analysing these experiments represents the first known attempt at quantifying building vulnerability to ballistics using fragility functions. The fragility and vulnerability functions developed from this research were then used in an impact assessment to determine likely building damage and insurance losses for a credible Auckland Volcanic Field (AVF) eruption scenario. The eruption modelled significant damage to buildings located within 3 km of the eruptive centre. In the ballistic impact assessment, over 800 ballistics directly impacted 486 buildings. Most impacts were modelled to have caused cladding perforation (maximum damage), but the overall damaged area on all impacted buildings was small enough that repairs are likely to be economic. In the tephra impact assessment over 200,000 buildings were impacted by >0.1 mm of ash with repair costs from structural damage estimated at $0.4-1.2 billion. However 97% of buildings impacted did not receive tephra loads sufficient to cause structural damage. Therefore non-structural damage is a large source of uncertainty in the impact assessment. The results from the literature review, building surveys, experiments and modelling have implications for response (e.g. life safety actions) and recovery planning (e.g. building repairs, building waste disposal) as part of disaster risk reduction in Auckland and globally.