Iosif L. Buchbinder graduated from Tomsk State University (Russia) with the highest degree (1988, doctor of ciences) in Physical and Mathematical ciences. From 1988 to the present time, he is a professor and head of Department of Theoretical Physics at Tomsk State Pedagogical University. He was a visiting professor at University of Pennsylvania, University of Maryland, University of North Carolina, University of Hannover, Humboldt University Berlin, Cambridge University, University of Sao Paulo, University of Juiz de Fora, University of Hiroshima, Wroclaw University, University of Brno, University of Beer Sheva. Was a supervisor of 21 Ph.D thesis, co-author of Effective Action in Quantum Gravity (IOPP, 1992; with S.D. Odintsov and I.L. Shapiro) and Ideas and Methods of Supersymmetry and Supergravity (IOPP, 1995, 1998; with S.M. Kuzenko), and author of around 300 papers in Statistical Mechanics, Quantum Field Theory, Quantum Gravity, Supersymmetry and Supergravity, and Higher Spin Field Theory.Ilya Shapiro graduated from Tomsk State University Physics (Russia) with the highest degree (1993, doctor of sciences) in Physical and Mathematical Sciences. In 1994-1996 was post-doc in the Department of Theoretical Physics in the University of Zaragoza, from 1996 visiting and from 2000 permanent professor at the Federal University of Juiz de Fora, Brazil. In 2014 he was visiting professor at the University of Geneva. Between 1985 and 2020 he supervised 13 Ph.D. theses, and has been author and co-author of 150+ articles and three books, including Effective Action In Quantum Gravity (IOPP, 1992), an introductory textbook on Newtonian Mechanics (two editions, in Brazil and in Germany), and an introductory textbook on Tensors and Relativity (Springer, 2019).

It is a good moment for summarizing the various advances, and these two authors are among the best experts in the specific field of quantum effective actions in gravity.

• PART I INTRODUCTION TO QUANTUM FIELD THEORY 1: Introduction 2: Relativistic Symmetry 3: Lagrange formalism in field theory 4: Field Models 5: Canonical Quantization of Free Fields 6: Scattering Matrix and Green Functions 7: Functional Integrals 8: Perturbation Theory 9: Renormalization 10: Quantum Gauge Theories PART II SEMICLASSICAL AND QUANTUM GRAVITY MODELS 11: Brief Review of General Relativity. 12: Classical fields in curved spacetime 13: Quantum fields in curved spacetime: renormalization 14: One-loop divergences 15: Renormalization group in curved space 16: Non-local form factors in flat and curved spacetime 17: Conformal anomaly and anomaly-induced action 18: General Notions of Perturbative Quantum Gravity 19: Massive ghosts in higher derivative models 20: One-loop renormalization in quantum gravity 21: Renormalization group in perturbative quantum gravity 22: Induced gravity approach 23: Final remarks on Part II References

Iosif L. Buchbinder graduated from Tomsk State University (Russia) with the highest degree (1988, doctor of ciences) in Physical and Mathematical ciences. From 1988 to the present time, he is a professor and head of Department of Theoretical Physics at Tomsk State Pedagogical University. He was a visiting professor at University of Pennsylvania, University of Maryland, University of North Carolina, University of Hannover, Humboldt University Berlin, Cambridge University, University of Sao Paulo, University of Juiz de Fora, University of Hiroshima, Wroclaw University, University of Brno, University of Beer Sheva. Was a supervisor of 21 Ph.D thesis, co-author of Effective Action in Quantum Gravity (IOPP, 1992; with S.D. Odintsov and I.L. Shapiro) and Ideas and Methods of Supersymmetry and Supergravity (IOPP, 1995, 1998; with S.M. Kuzenko), and author of around 300 papers in Statistical Mechanics, Quantum Field Theory, Quantum Gravity, Supersymmetry and Supergravity, and Higher Spin Field Theory.Ilya Shapiro graduated from Tomsk State University Physics (Russia) with the highest degree (1993, doctor of sciences) in Physical and Mathematical Sciences. In 1994-1996 was post-doc in the Department of Theoretical Physics in the University of Zaragoza, from 1996 visiting and from 2000 permanent professor at the Federal University of Juiz de Fora, Brazil. In 2014 he was visiting professor at the University of Geneva. Between 1985 and 2020 he supervised 13 Ph.D. theses, and has been author and co-author of 150+ articles and three books, including Effective Action In Quantum Gravity (IOPP, 1992), an introductory textbook on Newtonian Mechanics (two editions, in Brazil and in Germany), and an introductory textbook on Tensors and Relativity (Springer, 2019).

It is a good moment for summarizing the various advances, and these two authors are among the best experts in the specific field of quantum effective actions in gravity.

• PART I INTRODUCTION TO QUANTUM FIELD THEORY 1: Introduction 2: Relativistic Symmetry 3: Lagrange formalism in field theory 4: Field Models 5: Canonical Quantization of Free Fields 6: Scattering Matrix and Green Functions 7: Functional Integrals 8: Perturbation Theory 9: Renormalization 10: Quantum Gauge Theories PART II SEMICLASSICAL AND QUANTUM GRAVITY MODELS 11: Brief Review of General Relativity. 12: Classical fields in curved spacetime 13: Quantum fields in curved spacetime: renormalization 14: One-loop divergences 15: Renormalization group in curved space 16: Non-local form factors in flat and curved spacetime 17: Conformal anomaly and anomaly-induced action 18: General Notions of Perturbative Quantum Gravity 19: Massive ghosts in higher derivative models 20: One-loop renormalization in quantum gravity 21: Renormalization group in perturbative quantum gravity 22: Induced gravity approach 23: Final remarks on Part II References