  This paper deals with the class of Q-tensors, that is, a Q-tensor is a real
tensor $\mathcal{A}$ 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 subclasses
of Q-tensors are given: P-tensors, R-tensors, strictly semi-positive tensors
and semi-positive R$_0$-tensors. We prove that a nonnegative tensor is a
Q-tensor if and only if all of its principal diagonal entries are positive, and
so the equivalence of Q-tensor, R-tensors, strictly semi-positive tensors is
showed if they are nonnegative tensors. We also show that a tensor is a
R$_0$-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 if
and only if it is a R$_0$-tensor and the tensor complementarity problem $(\e,
\mathcal{A})$ has no non-zero vector solution, where $\e=(1,1\cdots,1)^\top$.
