ABSTRACT. These notes are based on an expository lecture that I gave at the workshop “Topology of Stratified Spaces” at MSRI Berkeley in September 2008. We will first explain the definition of a bordism invariant signature for a singular space, proceeding along a progression from less singular to more and more singular spaces, starting out from spaces that have no odd codimensional strata and, after having discussed Goresky—Siegel spaces andWitt spaces, ending up with general (non-Witt) stratified spaces. We will moreover discuss various refinements of the signature to orientation classes in suitable bordism theories based on singular cycles. For instance, we will indicate how one may define a symmetric 𝕃*-homology orientation for Goresky—Siegel spaces or a Sullivan orientation for those non-Witt spaces that still possess generalized Poincar´e duality. These classes can be thought of as refining the L-class of a singular space. Along the way, we will also see how to compute twisted versions of the signature and L-class.