  We present recent developments on the topic of the integrand reduction of
scattering amplitudes. Integrand-level methods allow to express an amplitude as
a linear combination of Master Integrals, by performing operations on the
corresponding integrands. This approach has already been successfully applied
and automated at one loop, and recently extended to higher loops. We describe a
coherent framework based on simple concepts of algebraic geometry, such as
multivariate polynomial division, which can be used in order to obtain the
integrand decomposition of any amplitude at any loop order. In the one-loop
case, we discuss an improved reduction algorithm, based on the application of
the Laurent series expansion to the integrands, which has been implemented in
the semi-numerical library Ninja. At two loops, we present the reduction of
five-point amplitudes in N=4 SYM, with a unitarity-based construction of the
integrand. We also describe the multi-loop divide-and-conquer approach, which
can always be used to find the integrand decomposition of any Feynman graph,
regardless of the form and the complexity of the integrand, with purely
algebraic operations.
