We investigate higher-order Weyl semimetals (HOWSMs) having bulk Weyl nodes attached to both surface and hinge Fermi arcs. We identify a new type of Weyl node, which we dub a 2nd-order Weyl node, that can be identified as a transition in momentum space in which both the Chern number and a higher order topological invariant change. As a proof of concept we use a model of stacked higher order quadrupole insulators (QI) to identify three types of WSM phases: 1st order, 2nd order, and hybrid order. The model can also realize type-II and hybrid-tilt WSMs with various surface and hinge arcs. After a comprehensive analysis of the topological properties of various HOWSMs, we turn to their physical implications that show the very distinct behavior of 2nd-order Weyl nodes when they are gapped out. We obtain three remarkable results: (i) the coupling of a 2nd-order Weyl phase with a conventional 1st-order one can lead to a hybrid-order topological insulator having coexisting surface cones and flat hinge arcs that are independent and not attached to each other. (ii) A nested 2nd-order inversion-symmetric WSM by a charge-density wave (CDW) order generates an insulating phase having coexisting flatband surface and hinge states all over the Brillouin zone. (iii) A CDW order in a time-reversal symmetric higher-order WSM gaps out a 2nd-order node with a 1st-order node and generates an insulating phase having coexisting surface Dirac cone and hinge arcs. Moreover, we show that a measurement of charge density in the presence of magnetic flux can help to identify some classes of 2nd-order WSMs. Finally, we show that periodic driving can be utilized as a way for generating HOWSMs. Our results are relevant to metamaterials as well as various phases of Cd_{3}As_{2}, KMgBi, and rutile-structure PtO_{2} that have been predicted to realize higher order Dirac semimetals.