We introduce Attribute-based Proxy Re-Signatures (ABPRS), an extension of Proxy Re-Signatures (PRS), which allows a semi-trusted proxy to transform a signature of one entity into a signature of another, without revealing any signing key and information about the signer. The proposed primitive combines features of PRS and Attribute-based Signatures (ABS), which is useful for many applications (we show how to apply ABPRS to three such applications). We propose a unidirectional ABPRS construction based on the hardness of the Inhomogeneous Small Integer Solution (ISIS) problem in standard lattices. We show that the proposed unidirectional ABPRS construction is multi-hop, where the transformation can be performed multiple times on any signature. Furthermore, we prove that the proposed unidirectional ABPRS construction satisfies the security notion of no-signature-query unforgeability and (perfect) privacy, under the ISIS assumption. Finally, we show how to obtain a multi-hop unidirectional ABPRS scheme that satisfies a stronger security notion of co-selective unforgeability by using a generic transformation technique.