This paper presents quantum amplitude amplification and estimation techniques, generalizing Grover's search algorithm and Shor's factoring algorithm. The key idea is to amplify the probability of finding a good solution in a quantum search problem. Given a Boolean function χ that partitions a set X into good and bad elements, and a quantum algorithm A that produces a superposition of elements of X, amplitude amplification allows finding a good x after an expected number of applications of A and its inverse proportional to 1/√a, where a is the probability of producing a good element. This provides a quadratic speedup over classical probabilistic methods. The paper also introduces amplitude estimation, a process that allows estimating the value of a. This is applied to the problem of approximate counting, where we wish to estimate the number of x in X such that χ(x) = 1. The quantum algorithm for approximate counting achieves optimal performance in various settings, providing an estimate within a specified error margin with high probability. The paper discusses the application of amplitude amplification to a broad class of classical heuristics, showing that it can provide quadratic speedup compared to any such heuristic. It also presents a quantum algorithm for amplitude estimation, which can be used to estimate the number of solutions to a Boolean function with high accuracy. The algorithm uses a number of function evaluations proportional to 1/√a, where a is the success probability of the quantum algorithm. The results show that quantum algorithms can achieve significant speedups over classical methods in search and counting problems.This paper presents quantum amplitude amplification and estimation techniques, generalizing Grover's search algorithm and Shor's factoring algorithm. The key idea is to amplify the probability of finding a good solution in a quantum search problem. Given a Boolean function χ that partitions a set X into good and bad elements, and a quantum algorithm A that produces a superposition of elements of X, amplitude amplification allows finding a good x after an expected number of applications of A and its inverse proportional to 1/√a, where a is the probability of producing a good element. This provides a quadratic speedup over classical probabilistic methods. The paper also introduces amplitude estimation, a process that allows estimating the value of a. This is applied to the problem of approximate counting, where we wish to estimate the number of x in X such that χ(x) = 1. The quantum algorithm for approximate counting achieves optimal performance in various settings, providing an estimate within a specified error margin with high probability. The paper discusses the application of amplitude amplification to a broad class of classical heuristics, showing that it can provide quadratic speedup compared to any such heuristic. It also presents a quantum algorithm for amplitude estimation, which can be used to estimate the number of solutions to a Boolean function with high accuracy. The algorithm uses a number of function evaluations proportional to 1/√a, where a is the success probability of the quantum algorithm. The results show that quantum algorithms can achieve significant speedups over classical methods in search and counting problems.