We present design methods for perfect reconstruction (PR) integer-modulated filterbanks, including biorthogonal (low-delay) filterbanks. Both the prototype filter and the modulation sequences are composed of integers, thus allowing efficient hardware implementations and fast computation. To derive such filterbanks, we first start with the PR conditions known for cosine modulation and extend them to more general, integer modulation schemes. For the design of biorthogonal PR integer prototypes, a lifting strategy is introduced. To find suitable integer modulation schemes, new algebraic methods are presented. We show solutions where the PR conditions on the prototype filters and the modulation matrices are entirely decoupled and where some simple coupling is introduced. Both even and odd numbers of channels are considered. Design examples are presented for both cases.