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For a graph <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>G</mi><mo>=</mo><mo stretchy="false">(</mo><mi>V</mi><mo>,</mo><mi>E</mi><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula> and a function <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>f</mi><mo>:</mo><mi>V</mi><mo>→</mo><mrow><mo>{</mo><mrow><mo>−</mo><mn>1</mn><mo>,</mo><mo>+</mo><mn>1</mn></mrow><mo>}</mo></mrow></mrow></semantics></math></inline-formula>, if <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>S</mi><mo>⊆</mo><mi>V</mi></mrow></semantics></math></inline-formula> then we write <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>S</mi><mo stretchy="false">)</mo><mo>=</mo><mstyle displaystyle="true"><munder><mo>∑</mo><mrow><mi>v</mi><mo>∈</mo><mi>S</mi></mrow></munder><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>v</mi><mo stretchy="false">)</mo></mrow></mstyle></mrow></semantics></math></inline-formula>. A function <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>f</mi></semantics></math></inline-formula> is said to be a non-zero vertex signed dominating function (for short, NVSDF) of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>G</mi></semantics></math></inline-formula> if <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>N</mi><mrow><mo>[</mo><mi>v</mi><mo>]</mo></mrow><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn></mrow></semantics></math></inline-formula> holds for every vertex <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>v</mi></semantics></math></inline-formula> in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>G</mi></semantics></math></inline-formula>, and the non-zero vertex signed domination number of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>G</mi></semantics></math></inline-formula> is defined as <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>γ</mi><mrow><mi>s</mi><mi>b</mi></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>=</mo><mi>max</mi><mrow><mo>{</mo><mrow><mi>f</mi><mrow><mo>(</mo><mi>V</mi><mo>)</mo></mrow><mrow><mo>|</mo><mrow><mrow><mi>f</mi><mo> </mo><mi>is</mi><mo> </mo><mi>an</mi><mo> </mo><mi>NVSDF</mi><mo> </mo><mi>of</mi><mo> </mo></mrow><mi>G</mi></mrow></mrow></mrow><mo>}</mo></mrow><mo>.</mo></mrow></semantics></math></inline-formula> In this paper, the novel concept of the non-zero vertex signed domination for graphs is introduced. There is also a special symmetry concept in graphs. Some upper bounds of the non-zero vertex signed domination number of a graph are given. The exact value of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>γ</mi><mrow><mi>s</mi><mi>b</mi></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> for several special classes of graphs is determined. Finally, we pose some open problems.