Exotic mesons are mesons that cannot be described as a quark-antiquark pair. The number of exotic mesons has been growing every year in the charm sector and the theoretical understanding of them is often conflicted amongst the community. Some possible explanations include hybrid mesons where the quark-antiquark pair is coupled to a gluonic excitation, compact tetraquarks where four quarks are bound into a localised state and molecules which consist of pairs of extended mesons. To study exotic mesons from first principles, lattice QCD provides the framework to perform spectroscopy calculations numerically. I will give a review and describe the relevant techniques used in this thesis.

After doing so, I will calculate masses of charmonium with angular momentum up to four. The results show QCD permits states with exotic quantum numbers that are not accessible by a quark-antiquark pair. I will identify states that are consistent with the quark-antiquark picture and then show that the remaining states in the extracted spectra can be interpreted to be the lightest and first excited hybrid meson supermultiplet.

Whilst the mass is one quantity that can be computed, hadron spectroscopy is also concerned with the calculation of the unstable properties of resonances which can decay into meson-meson states. These meson-meson states have four quarks and could also mix with tetraquarks. I will describe how to correctly extract the energies of four quark states within lattice QCD by reviewing operators resembling meson-mesons and then constructing a general class of operators resembling tetraquarks. I will then calculate a variety of spectra in the isospin-1 hidden charm sector and the doubly charmed sector. No evidence of a bound state or narrow resonance is found in these channels.

Having described how to include multi-meson states in lattice QCD, I will describe how to relate the lattice QCD spectrum to the scattering amplitudes and perform a calculation of elastic $DK$ scattering amplitudes which is relevant for the exotic $Ds0(2317)$. By analytically continuing the scattering amplitudes into the complex plane, I find a bound state pole near threshold which is in good agreement with what is found experimentally.