Gauge and gravitational anomalies are studied in two dimensions. In the gauge-anomaly case, we obtain the solution of the anomaly equation (the Wess-Zumino term) with only gauge fields, without auxiliary fields. Then we show, up to the second nontrivial order, that this solution agrees with the explicit Feynman-diagram calculation. In the gravitational-anomaly case we show, up to the second nontrivial order, that the anomaly obtained by the differential geometric method agrees with the explicit Feynman-diagram calculation.