With the development of high energy physics experiments, a large amount of exotic states in the hadronic sector have been observed. In order to shed some light on the nature of the tetraquark and pentaquark candidates, a constituent quark model, along with the Gaussian expansion method, has been employed systematically in real- and complex-range investigations. We review herein the double- and fully-heavy tetraquarks, but also the hidden-charm, hidden-bottom and doubly charmed pentaquarks. Several exotic hadrons observed experimentally were well reproduced within our approach; moreover, their possible compositeness and other properties, such as their decay widths and general patterns in the spectrum, are analyzed. Besides, we report also some theoretical predictions of tetra- and penta-quark states which have not seen by experiment yet.

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.

We use a color-magnetic interaction model (CMIM), a traditional constituent quark model and a multiquark color flux-tube model to systematically investigate the properties of the states $[Q_1{Q}_2][{\overline{Q}}_3{\overline{Q}}_4]$ ($Q=c$, $b$). The dynamical investigation indicates that the CMIM can not completely absorb QCD dynamical effects through the effective constituent quark mass and overestimates the color-magnetic interaction in the states under the assumption of the same spatial configurations. The Coulomb interaction plays a critical role in the dynamical model calculations on the heavy hadrons, which induces the fact that none of bound states $[Q_1{Q}_2][{\overline{Q}}_3{\overline{Q}}_4]$ can be found in the dynamical models. The color configuration ${[[Q_1{Q}_2{]}_{{\mathbf{6}}_c}[{\overline{Q}}_3{\overline{Q}}_4{]}_{{\overline{\mathbf{6}}}_c}]}_{\mathbf{1}}$ should be taken seriously in the ground states due to the strong Coulomb attraction between the $[Q_1{Q}_2{]}_{{\mathbf{6}}_c}$ and $[{\overline{Q}}_3{\overline{Q}}_4{]}_{{\overline{\mathbf{6}}}_c}$. The color configuration ${[[Q_1{Q}_2{]}_{{\overline{\mathbf{3}}}_c}[{\overline{Q}}_2{\overline{Q}}_4{]}_{{\mathbf{3}}_c}]}_{\mathbf{1}}$ is absolutely dominant in the excited states because of the strong Coulomb attraction within the $[Q_1{Q}_2{]}_{{\overline{\mathbf{3}}}_c}$ and $[{\overline{Q}}_2{\overline{Q}}_4{]}_{{\mathbf{3}}_c}$. The $J/\mathrm{\Psi }$-pair resonances recently observed by LHCb are difficult to be accommodated in the CMIM. The broad structure ranging from 6.2 to 6.8 GeV can be described as the ground tetraquark state $[cc][\overline{c}\overline{c}]$ in the various dynamical models. The narrow structure $X(6900)$ can be identified as the excited state $[cc][\overline{c}\overline{c}]$ with $L=1$ ($L=2$) in the constituent quark model (color flux-tube model).

Many proposals have been put forward to explore four-quark states $QQ\overline{q}\overline{q}$ ($Q=s$, $c$, $b$; $q=u$, $d$) by experiment, so a systematic study of $QQ\overline{q}\overline{q}$ spectrum with different constituent quark models by a high precision, few-body method, the Gaussian expression method, is useful. Three quark models: the Bhaduri, Cohler, Nogami quark model, the chiral quark model (ChQM), and the quark delocalization color screening model are all employed for a systematic calculation of the $S$-wave $QQ\overline{q}\overline{q}$ spectrum with different color structures, using the Gaussian expression method. The results show that only the $bb\overline{q}\overline{q}$ state with $(I,J)=(0,1)$ is bound in different color structures within the different quark models. The binding energy varies from several MeV for a di-meson structure to over 100 MeV for a diquark-antidiquark structure. For the $cc\overline{q}\overline{q}$ system, the state with $(I,J)=(0,1)$ is bound in a di-meson structure, and also bound in a diquark-antidiquark structure if pseudoscalar meson exchanges are accounted for. All are weakly bound states. The mixture of diquark-antidiquark and molecular structures is discussed in the framework of quark models for the first time; $cc\overline{q}\overline{q}$ with $(I,J)=(0,1)$ is below the threshold in addition to $bb\overline{q}\overline{q}$ in both the ChQM and the Bhaduri, Cohler, Nogami quark model. In the same channel, $ss\overline{q}\overline{q}$ is also a possible bound state with mass around 1.4 GeV in ChQM.

In the framework of the chiral quark model along with complex scaling range, we perform a dynamical study on the low-lying $S$-wave double-heavy tetraquark states ($QQ\overline{q}\overline{q}$, $Q=c$, $b$ and $q=u$, $d$) with an accurate computing approach, Gaussian expansion method. The meson-meson and diquark-antidiquark configurations within all possible color structures for spin-parity quantum numbers $J^P=0^{+}$, $1^{+}$, and $2^{+}$ and in the 0 and 1 isospin sectors are considered. Possible tightly bound and narrow resonance states are obtained for double-charm and double-bottom tetraquarks with $I{J}^P=01^{+}$, and these exotic states are also obtained in charm-bottom tetraquarks with $00^{+}$ and $01^{+}$ quantum numbers. Only a loosely bound state is found in charm-bottom tetraquarks of $02^{+}$ states. All of these bound states within meson-meson configurations are loosely bound whether in color-singlet channels or coupling to hidden-color ones. However, compact structures are available in diquark-antidiquark channels except for charm-bottom tetraquarks in $02^{+}$ states.