We consider the problem of distributed estimation, where a set of nodes is required to collectively estimate some parameter of interest from noisy measurements. The problem is useful in several contexts including wireless and sensor networks, where scalability, robustness, and low power consumption are desirable features. Diffusion cooperation schemes have been shown to provide good performance, robustness to node and link failure, and are amenable to distributed implementations. In this work we focus on diffusion-based adaptive solutions of the LMS type. We motivate and propose new versions of the diffusion LMS algorithm that outperform previous solutions. We provide performance and convergence analysis of the proposed algorithms, together with simulation results comparing with existing techniques. We also discuss optimization schemes to design the diffusion LMS weights.