| Summary: | In this paper, we give a characterization of normality of Toeplitz operator <inline-formula><math display="inline"><semantics><msub><mi>T</mi><mi>φ</mi></msub></semantics></math></inline-formula> on the Bergman space <inline-formula><math display="inline"><semantics><mrow><msup><mi>A</mi><mn>2</mn></msup><mrow><mo>(</mo><mi mathvariant="double-struck">D</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>. First, we state basic properties for Toeplitz operator <inline-formula><math display="inline"><semantics><msub><mi>T</mi><mi>φ</mi></msub></semantics></math></inline-formula> on <inline-formula><math display="inline"><semantics><mrow><msup><mi>A</mi><mn>2</mn></msup><mrow><mo>(</mo><mi mathvariant="double-struck">D</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>. Next, we consider the normal Toeplitz operator <inline-formula><math display="inline"><semantics><msub><mi>T</mi><mi>φ</mi></msub></semantics></math></inline-formula> on <inline-formula><math display="inline"><semantics><mrow><msup><mi>A</mi><mn>2</mn></msup><mrow><mo>(</mo><mi mathvariant="double-struck">D</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> in terms of harmonic symbols <inline-formula><math display="inline"><semantics><mi>φ</mi></semantics></math></inline-formula>. Finally, we characterize the normal Toeplitz operators <inline-formula><math display="inline"><semantics><msub><mi>T</mi><mi>φ</mi></msub></semantics></math></inline-formula> with non-harmonic symbols acting on <inline-formula><math display="inline"><semantics><mrow><msup><mi>A</mi><mn>2</mn></msup><mrow><mo>(</mo><mi mathvariant="double-struck">D</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>.
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