Theta series for lattice with indefinite signature (n$_+$,n$_-$) arise in different contexts of mathematics and physics. In a seminal work by Sander Zwegers in 2000, the modular properties for the n$_+$ = 1 case were understood to great details. On top of that it was expanded by Zagier-Bringmann-Ono. In a recent paper, we have understood how to extend that construction to the case of arbitrary signatures. In a few subsequent articles, Kudla, Westerholt-Raum , Bringmann et al, Nazaroglu reinterpreted our construction from various alternative directions, to which if time is permitted I would like to draw the attention of the audience. \par I will also talk about the appearance of such type of series in various interesting physics and mathematics settings. \\ 1) Compactification of N=2 type IIB string theory on Calabi-Yau threefolds , where D3-instantons are described in terms of a certain kind of mock-modular form. The partition function for D4-D2-D0 black holes for D4 instanton charge 2. \\ 2) In the context of moduli space of vector bundles on 4-manifolds: in favorable cases, one can consider the Donaldson invariant for the moduli space and that has a generating function which is a theta series of similar form. Such partition functions are also intimately related to the Vafa-Witten partition functions. \\ 3) These theta series are also closely connected to Appell-Lerch sums (or generalized Appell-Lerch sums).