It is usually assumed that interactions between individuals immediately affect the state of population. In reality, in biological models, results of interactions may appear in the future, and in social models, individuals or players may act, that is choose appropriate strategies, on the basis of the information concerning events in the past. It is well known that time delays may cause oscillations in dynamical systems. We will show that the presence of oscillations in such systems depends on particular causes of time delays. In particular, we will discuss two evolutionary game models with the same payoff matrix and with a stable and unstable interior stationary point. We modify above models to allow time delays to be strategy-dependent. They exhibit a novel behavior: after transient oscillations, the population settles at an equilibrium which depends on time delays. We will discuss stability of stationary states in stochastic models of finite populations with time delays.