The full-heavy tetraquarks $$bb{\bar{b}}{\bar{b}}$$b b b ¯ b ¯ and $$cc{\bar{c}}{\bar{c}}$$c c c ¯ c ¯ are systematically investigated within the chiral quark model and the quark delocalization color screening model. Two structures, meson–meson and diquark–antidiquark, are considered. For the full-beauty $$bb{\bar{b}}{\bar{b}}$$b b b ¯ b ¯ systems, there is no any bound state or resonance state in two structures in the chiral quark model, while the wide resonances with masses around $$19.1-19.4$$19.1 - 19.4 GeV and the quantum numbers $$J^{P}=0^{+}$$J P = 0 + , $$1^{+}$$1 + , and $$2^{+}$$2 + are possible in the quark delocalization color screening model. For the full-charm $$cc{\bar{c}}{\bar{c}}$$c c c ¯ c ¯ systems, the results are qualitative consistent in two quark models. No bound state can be found in the meson–meson configuration, while in the diquark–antidiquark configuration there may exist the resonance states, with masses range between 6.2 to 7.4 GeV, and the quantum numbers $$J^{P}=0^{+}$$J P = 0 + , $$1^{+}$$1 + , and $$2^{+}$$2 + . And the separation between the diquark and the antidiquark indicates that these states may be the compact resonance states. The reported state X(6900) is possible to be explained as a compact resonance state with $$IJ^{P}=00^{+}$$I J P = 00 + in present calculation. All these full-charm resonance states are worth searching in the experiments further.

In the framework of the color flux-tube model with a four-body confinement potential, the lowest charged tetraquark states $[Qq][{\overline{Q}}^{\prime }{\overline{q}}^{\prime }](Q=c,b,q=u,d,s)$ are studied by using the variational method, the Gaussian expansion method. The results indicate that some compact resonance states with three-dimensional spatial structures can be formed. These states cannot decay into two color singlet mesons $Q{\overline{q}}^{\prime }$ and ${\overline{Q}}^{\prime }q$ through the breakdown and recombination of color flux tubes but into $Q{\overline{Q}}^{\prime }$ and $q{\overline{q}}^{\prime }$. The four-body confinement potential is a crucial dynamical mechanism for the formation of these compact resonance states. The decay process is similar to that of a compound nucleus but due to the multibody color confinement. The newly observed charged states $Z_c(3900)$ and $Z_c(4025)/Z_c(4020)$ can be interpreted as the $S$-wave tetraquark states $[cu][\overline{c}\overline{d}]$ with quantum numbers $I{J}^P=11^{+}$ and $12^{+}$, respectively.

According to the classification of the quark model, the hadrons going beyond three-quark baryon and quark-antiquark meson pictures are called exotic hadrons. Many new hadrons have been observed since 2003, some of which exhibit exotic behaviors. There are a lot of excellent review articles on exotic hadrons available so far; the present article tries to focus on the recent experimental and theoretical progress on the exotic states from the perspective of the quark model. Although lattice quantum chromodynamics may give the final answer of the problem, the phenomenological models are still powerful tools to explore the exotic states and to provide insight on the phenomenology of hadrons. The spatial and color structures of multiquark states and the channel coupling calculation are emphasized through reviewing some bound states, molecular and color structure resonances. Finally, the unquench effects of some exotic states are reviewed. With the accumulation of experimental data on multiquark states and inspiration of underlying theory developments, more reasonable phenomenological models incorporating multi-body interactions and high Fock components to unify the description of normal hadrons and exotic hadrons are expectable.

In the framework of the quark delocalization color screening model, both the hidden-charm and hidden-bottom pentaquark resonances are studied in the hadron-hadron scattering process. A few narrow pentaquark resonances with a hidden-charm above 4.2 GeV and some narrow pentaquark resonances with a hidden-bottom above 11 GeV are found from corresponding scattering processes. In addition, the states $N{\eta }_c$, $NJ/\psi$, $N{\eta }_b$, and $N\mathrm{\Upsilon }$ with $I{J}^P=\frac{1}{2}{\frac{1}{2}}^{-}$ and $NJ/\psi$ and $N\mathrm{\Upsilon }$ with $I{J}^P=\frac{1}{2}{\frac{3}{2}}^{-}$ can all be bound by channel-coupling calculations. All of these heavy pentaquarks are worth investigating in future experiments.

Inspired by the present experimental results of charged charmonium-like states $Z_c^{+}$, we present a systematic study of the tetraquark states $[cu][\overline{c}\overline{d}]$ in a color flux-tube model with a multibody confinement potential. Our investigation indicates that charged charmonium-like states $Z_c^{+}(3900)$ or $Z_c^{+}(3885)$, $Z_c^{+}(3930)$, $Z_c^{+}(4020)$ or $Z_c^{+}(4025)$, $Z_1^{+}(4050)$, $Z_2^{+}(4250)$, and $Z_c^{+}(4200)$ can be described as a family of tetraquark $[cu][\overline{c}\overline{d}]$ states with the quantum numbers $n{{}^{2S+1}L}_J$ and $J^P$ of $1{{}^{3}S}_1$ and $1^{+}$, $2{{}^{3}S}_1$ and $1^{+}$, $1{{}^{5}S}_2$ and $2^{+}$, $1{{}^{3}P}_1$ and $1^{-}$, $1{{}^{5}D}_1$ and $1^{+}$, and $1{{}^{3}D}_1$ and $1^{+}$, respectively. The predicted lowest mass charged tetraquark state $[cu][\overline{c}\overline{d}]$ with $0^{+}$ and $1{{}^{1}S}_0$ lies at $3780\pm 10\mathrm{MeV}/{\mathrm{c}}^2$ in the model. These tetraquark states have compact three-dimensional spatial configurations similar to a rugby ball with higher orbital angular momentum $L$ between the diquark $[cu]$ and antidiquark $[\overline{c}\overline{d}]$ corresponding to a more prolate spatial distribution. The multibody color flux tube, a collective degree of freedom, plays an important role in the formation of those charged tetraquark states. However, the two heavier charged states $Z_c^{+}(4430)$ and $Z_c^{+}(4475)$ cannot be explained as tetraquark states $[cu][\overline{c}\overline{d}]$ in this model approach.

The fully heavy pentaquarks $cccc\overline{c}$ and $bbbb\overline{b}$ are systematically investigated within the chiral quark model and quark delocalization color screening model. The results are consistent in both of the models, and the effect of the channel coupling is crucial for forming a bound state of the fully heavy pentaquark system. Three possible fully heavy pentaquarks are obtained and are the $cccc\overline{c}$ state with $J^P=1/2^{-}$ and the mass of 7891.9–7892.7 MeV, the $bbbb\overline{b}$ state with $J^P=1/2^{-}$ and the mass of 23810.1–23813.8 MeV, and the $bbbb\overline{b}$ state with $J^P=3/2^{-}$ and the mass of 23748.2–23752.3 MeV. All these fully heavy pentaquark states are worth searching in future experiments.