This text discusses the wave front set of pseudodifferential operators and its generalizations. Theorem 3.8 states that if a segment γ in the cotangent bundle does not intersect the wave front set of Pu, then γ either lies entirely within the wave front set of u or does not intersect it. Generalizations of the wave front set, such as the analytic wave front and Gevrey wave front, are also discussed. The concept of the wave front set for distributions in Sobolev spaces is introduced, and Theorem 3.9 shows that the wave front set of a distribution is contained within the characteristic set of a pseudodifferential operator or the wave front set of the operator applied to the distribution. Theorem 3.10 extends this result under certain conditions. Section 4 introduces Fourier integral operators, which are a broader class of operators than pseudodifferential operators. These operators are defined by an integral involving an amplitude and a phase function. Examples include pseudodifferential operators and solutions to hyperbolic equations, which can be expressed using Fourier integral operators. The text also provides an example of a Cauchy problem for the wave equation, which is solved using two Fourier integral operators and a smooth kernel operator. Another example involves a first-order pseudodifferential equation, where the parametrix is given by a Fourier integral operator.This text discusses the wave front set of pseudodifferential operators and its generalizations. Theorem 3.8 states that if a segment γ in the cotangent bundle does not intersect the wave front set of Pu, then γ either lies entirely within the wave front set of u or does not intersect it. Generalizations of the wave front set, such as the analytic wave front and Gevrey wave front, are also discussed. The concept of the wave front set for distributions in Sobolev spaces is introduced, and Theorem 3.9 shows that the wave front set of a distribution is contained within the characteristic set of a pseudodifferential operator or the wave front set of the operator applied to the distribution. Theorem 3.10 extends this result under certain conditions. Section 4 introduces Fourier integral operators, which are a broader class of operators than pseudodifferential operators. These operators are defined by an integral involving an amplitude and a phase function. Examples include pseudodifferential operators and solutions to hyperbolic equations, which can be expressed using Fourier integral operators. The text also provides an example of a Cauchy problem for the wave equation, which is solved using two Fourier integral operators and a smooth kernel operator. Another example involves a first-order pseudodifferential equation, where the parametrix is given by a Fourier integral operator.