We study continuous-time nonlinear systems, first in the case where there are continuous output measurements and next in the case where there are only discrete output measurements. When continuous measurements are available, we provide observers that converge in finite time. When only discrete measurements are available, we provide observers that do not converge in finite time, but which do converge asymptotically with a rate of convergence that is proportional to the negative of the logarithm of the size of the sampling interval. We illustrate our results in a pendulum example.