Neural methods for amortized inference have emerged as a powerful approach to statistical inference, leveraging the representational capacity of neural networks, optimization libraries, and graphics processing units to learn complex mappings between data and inferential targets. These methods allow for rapid inference after an initial setup cost, as they utilize fast feed-forward operations. The article reviews recent progress in amortized inference, covering point estimation, approximate Bayesian inference, summary-statistic construction, and likelihood approximation. It also discusses software tools and provides a simple illustration of the benefits of amortized inference over Markov chain Monte Carlo (MCMC) methods. The article concludes with an overview of relevant topics and future research directions. Amortized inference is defined as a process where the initial cost of training a neural network is "amortized over time" as the trained network is reused for inferences with new data. This concept is analogous to how humans reuse past inferences to make quick decisions. Neural networks are particularly suited for tasks requiring repeated complex computations, as the initial training cost can be offset by the efficiency of subsequent inferences. The article discusses amortization from a decision-theoretic perspective, emphasizing the optimization of decision rules through the minimization of expected losses or KL divergences. It covers neural Bayes estimators, which approximate Bayesian posterior distributions by minimizing the KL divergence between the true posterior and an approximate distribution. The article also explores methods for approximate Bayesian inference via KL-divergence minimization, including forward and reverse KL divergence approaches. Neural networks are used to construct summary statistics, which are essential in simulation-based inference. These methods can be explicit, where summary statistics are directly constructed, or implicit, where summary statistics are learned through the network's architecture. The number of summary statistics is a design choice, often determined by the dimensionality of the data and the complexity of the model. The article also discusses neural methods for amortized likelihood-function and likelihood-to-evidence-ratio approximation. These methods enable the efficient computation of likelihoods and likelihood-to-evidence ratios, which are crucial for hypothesis testing, model comparison, and MCMC algorithms. Amortized likelihood approximators can be used for both frequentist and Bayesian inference with multiple conditionally i.i.d. replicates. Overall, the article highlights the advantages of neural amortized inference, including faster computation, reduced computational burden, and the ability to handle complex models and large datasets. It also discusses the challenges and limitations, such as the amortization gap, which arises from incomplete training or neural network inflexibility. The article concludes with an outlook on future research directions in this rapidly evolving field.Neural methods for amortized inference have emerged as a powerful approach to statistical inference, leveraging the representational capacity of neural networks, optimization libraries, and graphics processing units to learn complex mappings between data and inferential targets. These methods allow for rapid inference after an initial setup cost, as they utilize fast feed-forward operations. The article reviews recent progress in amortized inference, covering point estimation, approximate Bayesian inference, summary-statistic construction, and likelihood approximation. It also discusses software tools and provides a simple illustration of the benefits of amortized inference over Markov chain Monte Carlo (MCMC) methods. The article concludes with an overview of relevant topics and future research directions. Amortized inference is defined as a process where the initial cost of training a neural network is "amortized over time" as the trained network is reused for inferences with new data. This concept is analogous to how humans reuse past inferences to make quick decisions. Neural networks are particularly suited for tasks requiring repeated complex computations, as the initial training cost can be offset by the efficiency of subsequent inferences. The article discusses amortization from a decision-theoretic perspective, emphasizing the optimization of decision rules through the minimization of expected losses or KL divergences. It covers neural Bayes estimators, which approximate Bayesian posterior distributions by minimizing the KL divergence between the true posterior and an approximate distribution. The article also explores methods for approximate Bayesian inference via KL-divergence minimization, including forward and reverse KL divergence approaches. Neural networks are used to construct summary statistics, which are essential in simulation-based inference. These methods can be explicit, where summary statistics are directly constructed, or implicit, where summary statistics are learned through the network's architecture. The number of summary statistics is a design choice, often determined by the dimensionality of the data and the complexity of the model. The article also discusses neural methods for amortized likelihood-function and likelihood-to-evidence-ratio approximation. These methods enable the efficient computation of likelihoods and likelihood-to-evidence ratios, which are crucial for hypothesis testing, model comparison, and MCMC algorithms. Amortized likelihood approximators can be used for both frequentist and Bayesian inference with multiple conditionally i.i.d. replicates. Overall, the article highlights the advantages of neural amortized inference, including faster computation, reduced computational burden, and the ability to handle complex models and large datasets. It also discusses the challenges and limitations, such as the amortization gap, which arises from incomplete training or neural network inflexibility. The article concludes with an outlook on future research directions in this rapidly evolving field.