We provide new classes of finite-time and fastconverging observers for continuous-time nonlinear systems that contain unknown constant parameters and measurement delays. When the dynamics are affine in the unmeasured state, we obtain fixed time observers for states and for the model parameters, i.e., the finite convergence time is independent of the initial state of the dynamics. When the dynamics contain a nonlinearity that depends on the unmeasured state, we instead obtain asymptotic convergence of our observers, whose rate of convergence converges to infinity as the growth rate of this nonlinearity converges to zero, so in this case we call the observers almost finite-time observers. Our examples illustrate the ease with which our assumptions are verified in practice.