These are the expanded notes of a mini-course of four lectures by the same title given in the workshop "p-adic aspects of modular forms" held at IISER Pune, in June, 2014. We give a brief introduction of p-adic L-functions attached to certain types of automorphic forms on GLn with the specific aim to understand the p-adic symmetric cube L-function attached to cusp forms on GL 2 over rational numbers. Contents 1. What is a p-adic L-function? 2 2. The symmetric power L-functions 11 3. p-adic L-functions for GL 4 16 4. p-adic L-functions for GL 3 × GL 2 22 References 27The aim of this survey article is to bring together some known constructions of the p-adic L-functions associated to cohomological, cuspidal automorphic representations on GL n /Q. In particular, we wish to briefly recall the various approaches to construct p-adic L-functions with a focus on the construction of the p-adic L-functions for the Sym 3 transfer of a cuspidal automorphic representation π of GL 2 /Q. We note that p-adic L-functions for modular forms or automorphic representations are defined using p-adic measures. In almost all cases, these p-adic measures are constructed using the fact that the L-functions have integral representations, for example as suitable Mellin transforms. Candidates for distributions corresponding to automorphic forms can be written down using such integral representations of the L-functions at the critical points. The well-known Prop. 2 is often used to prove that they are indeed distributions, which is usually a consequence of the defining relations of the Hecke operators. Boundedness of these distributions are shown by proving certain finiteness or integrality properties, giving the sought after p-adic measures.In Sect. 1, we discuss general notions concerning p-adic L-functions, including our working definition of what we mean by a p-adic L-function. As a concrete example, we discuss the construction of the p-adic L-functions that interpolate critical values of L-functions attached to modular forms. Manin [47]