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.

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.

The constituent quark model is used to compute the ground and excited state masses of QQQbaryons containing either c or bquarks. The quark model parameters previously used to describe the properties of charmonium and bottomonium states were used in this analysis. The non-relativistic three-body bound state problem is solved by means of the Gaussian expansion method which provides sufficient accuracy and simplifies the subsequent evaluation of the matrix elements. Several low-lying states with quantum numbers and are reported. We compare the results with those obtained by the other theoretical formalisms. There is a general agreement for the mass of the ground state in each sector of triply heavy baryons. However, the situation is more puzzling for the excited states, and appropriate comments about the most relevant features of our comparison are given.

The recent experimental results of the LHCb Collaboration suggested the existence of pentaquark states with a charmonium. To understand the structure of the states, a dynamical calculation of 5-quark systems with quantum numbers $I{J}^P=\frac{1}{2}(\frac{1}{2}{)}^{\pm }$, $\frac{1}{2}(\frac{3}{2}{)}^{\pm }$ and $\frac{1}{2}(\frac{5}{2}{)}^{\pm }$ is performed in the framework of the chiral quark model with the help of the Gaussian expansion method. The results show that there are several negative parity resonance states while all of the positive parity states are the scattering states. The $P_c(4380)$ state is suggested to be the pentaquark state of ${\mathrm{\Sigma }}_c^{*}\overline{D}$. Although the energy of ${\mathrm{\Sigma }}_c{\overline{D}}^{*}$ is very close to the mass of $P_c(4450)$, the inconsistent parity prevents the assignment. The calculated distances between quarks confirm the molecular nature of the states.

Fully-heavy tetraquark states, i.e., $cc\overline{c}\overline{c}$, $bb\overline{b}\overline{b}$, $bb\overline{c}\overline{c}$ ($cc\overline{b}\overline{b}$), $cb\overline{c}\overline{c}$, $cb\overline{b}\overline{b}$, and $cb\overline{c}\overline{b}$, are systematically investigated by means of a nonrelativistic quark model based on lattice-QCD studies of the two-body $Q\overline{Q}$ interaction, which exhibits a spin-independent Cornell potential along with a spin-spin term. The four-body problem is solved using the Gaussian expansion method; additionally, the so-called complex scaling technique is employed so that bound, resonance, and scattering states can be treated on the same footing. Moreover, a complete set of four-body configurations, including meson-meson, diquark-antidiquark, and K-type configurations, as well as their couplings, are considered for spin-parity quantum numbers $J^{P(C)}=0^{+(+)}$, $1^{+(\pm )}$, and $2^{+(+)}$ in the $S$-wave channel. Several narrow resonances, with two-meson strong decay widths less than 30 MeV, are found in all of the tetraquark systems studied. Particularly, the fully-charm resonances recently reported by the LHCb Collaboration, at the energy range between 6.2 and 7.2 GeV in the di-$J/\psi$ invariant spectrum, can be well identified in our calculation. Focusing on the fully-bottom tetraquark spectrum, resonances with masses between 18.9 and 19.6 GeV are found. For the remaining charm-bottom cases, the masses are obtained within a energy region from 9.8 GeV to 16.4 GeV. All these predicted resonances can be further examined in future experiments.

The LHCb Collaboration has recently reported strong evidences of the existence of pentaquark states in the hidden-charm baryon sector, the so-called $P_c(4380{)}^{+}$ and $P_c(4450{)}^{+}$ signals. Five-quark bound states in the hidden-charm sector were explored by us using, for the quark-quark interaction, a chiral quark model which successfully explains meson and baryon phenomenology, from the light to the heavy quark sector. We extend herein such study into the hidden-bottom pentaquark sector, analyzing possible bound-states with spin-parity quantum numbers $J^P={\frac{1}{2}}^{\pm }$, ${\frac{3}{2}}^{\pm }$ and ${\frac{5}{2}}^{\pm }$, and in the $\frac{1}{2}$ and $\frac{3}{2}$ isospin sectors. We do not find positive parity hidden-bottom pentaquark states; however, several candidates with negative parity are found with dominant baryon-meson structures ${\mathrm{\Sigma }}_b^{(*)}{\overline{B}}^{(*)}$. Their inner structures have been also analyzed with the computation of the distance among any pair of quarks within the bound-state. This exercise reflects that molecular-type bound-states are favored when only color-singlet configurations are considered in the coupled-channels calculation whereas some deeply-bound compact pentaquarks can be found when hidden-color configurations are added. Finally, our findings resemble the ones found in the hidden-charm sector but, as expected, we find in the hidden-bottom sector larger binding energies and bigger contributions of the hidden-color configurations.