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This paper deals with the class of Q-tensors, that is, a Q-tensor is a realtensor $\mathcal{A}$ such that the tensor complementarity problem $(\q,\mathcal{A})$:  $$\mbox{ finding } \x \in \mathbb{R}^n\mbox{ such that }\x \geq \0, \q +\mathcal{A}\x^{m-1} \geq \0, \mbox{ and }\x^\top (\q + \mathcal{A}\x^{m-1}) =0, $$ has a solution for each vector $\q \in \mathbb{R}^n$. Several subclassesof Q-tensors are given: P-tensors, R-tensors, strictly semi-positive tensorsand semi-positive R$_0$-tensors. We prove that a nonnegative tensor is aQ-tensor if and only if all of its principal diagonal entries are positive, andso the equivalence of Q-tensor, R-tensors, strictly semi-positive tensors isshowed if they are nonnegative tensors. We also show that a tensor is aR$_0$-tensor if and only if the tensor complementarity problem $(\0,\mathcal{A})$ has no non-zero vector solution, and a tensor is a R-tensor ifand only if it is a R$_0$-tensor and the tensor complementarity problem $(\e,\mathcal{A})$ has no non-zero vector solution, where $\e=(1,1\cdots,1)^\top$.

Properties of Tensor Complementarity Problem and Some Classes of Structured Tensors