This paper studies the impact of service level constraints in the context of the capacitated and uncapacitated lot sizing problems with deterministic demand and backlogging over a discrete and finite time horizon. Because the standard formulation of these problems cannot distinguish between a backlog and a backorder, we use a reformulation to explicitly make this distinction. We then formulate several service level constraints adapted from the stochastic inventory control literature and introduce new ones. We analyze the impact of these constraints on the structure of the optimal solutions in terms of the number of backorders and total backlog, and on the performance of these solutions in terms of various service level indicators (proportion of periods with no backlog, proportion of backorders over total demand, maximum delay, average unit waiting time for an item). We also consider the introduction of various types of backordering costs (both fixed and duration dependent) and study their impact. Finally, we analyze the impact of a first-in first-out order management policy and show that it helps improve most solutions.