Employee timetabling is the operation of assigning employees to tasks in a set of shifts during a fixed period of time, typically a week. We present a general definition of employee timetabling problems (ETPs) that captures many real-world problem formulations and includes complex constraints. The proposed model of ETPs can be represented in a tabular form that is both intuitive and efficient for constraint representation and processing. The constraint networks of ETPs include non-binary constraints and are difficult to formulate in terms of simple constraint solvers. We investigate the use of local search techniques for solving ETPs. In particular, we propose several versions of hill-climbing that make use of a novel search space that includes also partial assignments. We show that, on large and difficult instances of real world ETPs, where systematic search fails, local search methods perform well and solve the hardest instances. According to our experimental results on various techniques, a simple version of hill climbing based on random moves is the best method for solving large ETP instances.