The field of Tiny Machine Learning (TinyML) has gained significant attention due to its potential to enable intelligent applications on resource-constrained devices. This review provides an in-depth analysis of the advancements in efficient neural networks and the deployment of deep learning models on ultra-low-power microcontrollers (MCUs) for TinyML applications. It introduces neural networks and discusses their architectures and resource requirements. It explores MEMS-based applications on ultra-low-power MCUs, highlighting their potential for enabling TinyML on resource-constrained devices. The review focuses on efficient neural networks for TinyML. It covers techniques such as model compression, quantization, and low-rank factorization, which optimize neural network architectures for minimal resource utilization on MCUs. The article then delves into the deployment of deep learning models on ultra-low-power MCUs, addressing challenges such as limited computational capabilities and memory resources. Techniques such as model pruning, hardware acceleration, and algorithm-architecture co-design are discussed. Lastly, the review provides an overview of current limitations in the field, including the tradeoff between model complexity and resource constraints. Overall, this review article presents a comprehensive analysis of efficient neural networks and deployment strategies for TinyML on ultra-low-power MCUs. It identifies future research directions for unlocking the potential of TinyML applications on resource-constrained devices.