Transition-metal dichalcogenides (TMDs) are renowned for their rich and varied bulk properties, while their single-layer variants have become one of the most prominent examples of two-dimensional materials beyond graphene. Their disparate ground states largely depend on transition metal d-electron-derived electronic states, on which the vast majority of attention has been concentrated to date. Here, we focus on the chalcogen-derived states. From density-functional theory calculations together with spin-and angle-resolved photoemission, we find that these generically host a coexistence of type-I and type-II three-dimensional bulk Dirac fermions as well as ladders of topological surface states and surface resonances. We demonstrate how these naturally arise within a single p-orbital manifold as a general consequence of a trigonal crystal field, and as such can be expected across a large number of compounds. Already, we demonstrate their existence in six separate TMDs, opening routes to tune, and ultimately exploit, their topological physics.The classification of electronic structures based on their topological properties has opened powerful routes for understanding solid state materials. 1 The nowfamiliar Z 2 topological insulators are most renowned for their spin-polarised Dirac surface states residing in inverted bulk band gaps. 1 In systems with rotational invariance, a band inversion on the rotation axis can generate protected Dirac cones with a point-like Fermi surface of the bulk electronic structure. 2-8 If either inversion or time-reversal symmetry is broken, a bulk Dirac point can split into a pair of spin-polarised Weyl points. 9-15 Unlike for elementary particles, Lorentz-violating Weyl fermions can also exist in the solid state, manifested as a tilting of the Weyl cone. If this tilt is sufficiently large, so-called type-II Weyl points can occur, now formed at the touching of open electron and hole pockets. [15][16][17][18][19][20][21][22] Realising such phases in solid-state materials not only offers unique environments and opportunities for studying the fundamental properties of fermions, but also holds potential for applications exploiting their exotic surface excitations and bulk electric and thermal transport properties. [23][24][25][26][27] Consequently, there is an intense current effort focused on identifying compounds which host the requisite band inversions. In many cases, however, this arXiv:1702.08177v2 [cond-mat.mtrl-sci]