Many concepts from solid state physics may be applied to electromagnetic waves propagating in periodic media. A large class of materials, called photonic crystals, that to some measure mimic natural materials, have been extensively studied since the end of the 20th century. In the last decade, after the isolation of the first two-dimensional material, graphene, interest in the artificial graphene systems has emerged. This work presents the results of mimicking graphene with the two microwave artificial analogues. The simplicity of the fabrication process and measurement techniques means they are perfect candidates for studying physics in natural graphene. By measuring near-fields across the samples we not only obtain local distribution of the electric fields, but are also able to plot the experimental dispersion relationships that are lacking in previous studies. The first artificial graphene comprised of cylindrical metallic rods, which replicate carbon atoms, was fabricated and characterised. Dispersion curves of the bound electromagnetic eigenmodes were experimentally determined by measuring the electric near-fields just above the surface. Two linear crossings are evident in these dispersion curves at each of the K and K' points of the Brillouin zone, mimicking the well-celebrated Dirac cones in real graphene. Breaking inversion symmetry of the system, which leads to the opening of the band gap, is also demonstrated in this work. The second structure with a smaller ratio of out-of-plane to in-plane dimensions, more akin to real graphene, is  comprised of in-plane metallic wires forming a hexagonal mesh. In this configuration, metal wires replicate bonding terms between the carbon atoms in graphene. Like the first structure, it features gapless Dirac dispersion at the corners of the Brillouin zone. We propose a simple, analytical LC circuit model capable of representing the electrodynamics of propagating modes inside the hexagonal mesh. Dispersion curves calculated with this circuit model are shown to fully match the experimental data using realistic values of the inductance and capacitance of the wire mesh. We suggest and show experimentally that by modifying wires individually one can introduce an effect similar to straining of the graphene. The latter structure was used for studying topological edge modes supported at the interface between the two oppositely modified structures. A new super-cell of the hexagonal wire mesh lattice results in a double Dirac cone at the Γ point. Contraction and expansion of the hexagon in the super-cell open the trivial and non-trivial band gaps. Edge modes that exist at the interface of the expanded and contracted structures are studied. We reveal that the direction of the interface may or may not protect the edge modes from being hybridised. By scanning and measuring the near-field across the entire sample comprised of the two modified structures we obtain dispersion relationships for both surface and edge modes simultaneously. The manifestation of the edge modes hybridisation is shown in measured near-field distributions and supported by the analytical LC model and the effective Hamiltonian description.Engineering and Physical Sciences Research Council (EPSRC)