This paper proposes models and algorithms for the pickup and delivery vehicle routing problem with time windows and last-in-first-out (LIFO) loading constraints (PDPTWL). The LIFO loading rule ensures that no handling is required prior to unloading an item from the vehicle: a linear stack loading structure is maintained and an item can only be delivered if it is the last one in the stack. Three exact branch-price-and-cut algorithms are proposed for this problem. The first incorporates the LIFO constraints in the master problem. The second one handles the LIFO constraints directly in the shortest path pricing problem. It applies a dynamic programming algorithm relying on an ad hoc dominance criterion. The third algorithm is a hybrid between the first two methods. Known valid inequalities are adapted to the PDPTWL and the impact of different path relaxations on the total computation time is investigated. Computational results obtained on instances derived from known PDPTW instances are reported.