We start by defining the Grothendieck ring of varieties and the Burnside ring\, discussing their fundamental properties and open questions. Then we explain various types of motivic invariants of birational maps along with their basic properties\, including the more refined horizontal and vertical motivic invariants for maps between fibrations. We state vanishing results of motivic invariants for surfaces over perfect fields and prove vanishing for all threefolds over the field of complex numbers\, using the intermediate Jacobians and MRC fibrations. Finally\, we explain the nonvanishing of motivic invariants for various Cremona groups\, starting with P^3. For P^4 over the field of complex numbers\, we explain the unboundedness of motivic invariants\, via K3 and elliptic surfaces. Time permitting\, we also introduce invariants of pairs that relate motivic invariants to the construction of Genevois-Lonjou-Urech.