This paper is concerned with the study of the large-time behavior of the solutions u of a class of one-dimensional reaction-diffusion equations with monostable reaction terms f , including in particular the classical Fisher-KPP nonlinearities. The nonnegative initial data u_0(x) are chiefly assumed to be exponentially bounded as x tends to +∞ and separated away from the unstable steady state 0 as x tends to −∞. On the one hand, we give some conditions on u_0 which guarantee that, for some λ > 0, the quantity c_λ = λ + f (0)/λ is the asymptotic spreading speed, in the sense that lim t→+∞ u(t, ct) = 1 (the stable steady state) if c < c_λ and limt→+∞ u(t, ct) = 0 if c > c_λ . These conditions are fulfilled in particular when u_0(x) e^{λx} is asymptotically periodic as x → +∞. On the other hand, we also construct examples where the spreading speed is not uniquely determined. Namely, we show the existence of classes of initial conditions u_0 for which the ω−limit set of u(t, ct + x) as t tends to +∞ is equal to the whole interval [0, 1] for all x ∈ R and for all speeds c belonging to a given interval (γ1 , γ2 ) with large enough γ1 < γ2 .