These lectures are devoted to semiclassical spectral asymptotics with accurate remainder estimates and their applications to spectral asymptotics of other types. In O. Introduction, the brief description of the hyperbolic operator method is given. In I. Why one should study local semi classical spectral asymptotics ? we show how starting from rather classical theorems concerning LSSA (local semi classical spectral asymptotics) one can weaken their conditions. In 2. How local semi classical spectral asymptotics yield standard spectral asymptotics ? we show how LSSA yield asymptotics of eigenvalues tending to $+\infty $ for operators on compact manifolds and for operators on $\mathbf {R}^d$ with potentials increasing at infinity and asymptotics of eigenvalues tending to $-O$ for operators in $\mathbf {R}^d$ with potentials decreasing at infinity. In 3 How can one derive local semi classical spectral asymptotics in the general case ?, we present basic ideas permitting us to use the hyperbolic operator method for general matrix operators and for operators on manifolds with boundary. In 4. Propagation of singularities, we apply the short-time propagation of singularities in order to justify the previous section construction ; then in 5. Tauberian theorem, we derive LSSA. We treat the long-time propagation of singularities in order to improve the remainder estimate in LSSA in 6. How to improve remainder estimate in the case of non periodic trajectories ? and in 7. How to improve remainder estimate in the case of periodic trajectories ? In the last case the final formula contains non-Weylian term. In 8. Eigenvalue estimates and asymptotics for spectral problems with singularities, we split LSSA and Lieb-Cwickel-Rozenbljum eigenvalue estimate and derive estimates above and below for the number of the eigenvalues for the Schrödinger operator. Taking this operator depending on some parameters we obtain asymptotics with respect to this parameter. In 9. Generalizations. Non-Weylian asymptotics, more advanced development of the theory is presented.