Thermal and Moisture Protection, Roofing, and Waterproofing
Heat, air, and vapor
R-value is thermal resistance (higher is better); U-value = 1/R. Insulation must be continuous to defeat thermal bridging at studs and slab edges. The vapor retarder goes on the warm side - interior in heating climates - and an air barrier is a separate function; do not let a question merge them. Where the dew point falls inside the assembly, condensation and mold follow.
Roofing
Low-slope generally means 3:12 or less, and most low-slope systems need positive slope of at least 1/4 in per foot to drains or scuppers. Systems: built-up roofing, modified bitumen, EPDM, TPO, PVC, and standing-seam metal. Steep-slope asphalt shingles typically require 4:12; between 2:12 and 4:12 most manufacturers require double underlayment. Ice-dam membrane at eaves is standard in cold climates. Details fail more often than membranes: base flashing, counterflashing, crickets behind chimneys, drain and scupper heights, and terminations.
Waterproofing and below-grade
Dampproofing resists soil moisture only; waterproofing resists hydrostatic head. Below-grade options include bentonite, sheet membrane, and fluid-applied with reinforcing fabric, always protected by drainage board and free-draining backfill. Positive-side (exterior) placement is preferred because it keeps water out of the assembly. Above grade, use a weather-resistive barrier with shingle-lapped seams.
Worked example
A roof 100 ft long draining to one edge at 1/4 in per foot: total fall = 100 x 0.25 = 25 in, roughly 2 ft 1 in. Confirm the drain or scupper rim sits at least that far below the high point, and check crickets at penetrations.
Traps: vapor retarder on the cold side, dampproofing offered as waterproofing, and claiming underlayment makes a roof watertight.