Warning: The pdf version contains mistakes and typos that have been corrected in the printed version. Here, only those in the printed version are listed.

Typos:
1. p54, Renark 3.10: The explicit orthogonal transformation is the identity for b=0 and, therefore, does not in this case prove the fact that X has CM. But it is easy to write down an explicit example in this case as well.

2. p54,l-9: The notation is not very clear at this point: The Galois groups acts on the etale cohomolo\ gy of the base change of $X$ to an algebraic closure $\bar X=X\times_k\bar k$.

3. p29. Example 3.11: The map given to describe the elliptic fibration is to be understood as a rational map. On the other chart, the map is given by the other factors of the two sums. Also, the first component should be $x_0^2+\xi^2x_1^2$. (Thanks to Oliver Roendigs.)

4. p73, Proposition 3.5: On the left hand side it should read KS(X) (and not KS(T(X)). The dimension would just not fit otherwise.

5. p75, line before Remark 4.5: It should read b_2(Y_0\times\bar k)>3$6. p81, footnote: The quotient should by$Pic(T')$(and not$Pic(T)$) 7. p82, l11: with the scheme$M_d$8. p82, l21:$\pi\colon M_d\to N$9. p101, Remark 4.8: Instead of$\Lambda/\ell\Lambda$it should be$(\Lambda/\ell\Lambda)^{\oplus 22}$. 10. p196: The reference for Theorem 5.4 is [143, Thm. 1.6]. 11. p219, l-3: The reference should be [236, III.Prop.9.7] 12. p220, l14: In equation for Fermat quartic also$x_0$should be raised to the forth power. 13. p259, penultimate sentence of proof of Cor. 1.5: The injectivity does not hold in general, but it does hold for K3 surfaces. (Thanks to Alvaro Muniz Brea for pointing this out.) 14. p314: There is a sign mistake in the definition of the ring structure. The last summand should be$-(-\lambda\mu'+\lambda'\mu+(x.y))f$. (Thanks to Emma Brakkee for pointing this out to me.) 15. p323, l-6: quartic(!) surface 16. p329, l-10: The reference to Bertini's work is wrong. It should be his article Reseaux de Kummer et surfaces K3. Invent. Math. 93 (1988), 267--284. (Thanks to C. Peters for pointing this out.) 17. p336, Corollary 1.14:$\varphi(m)$is of course not the order of$\mu_m$, but the order of the Galois group of the corresponding cyclotomic field. 18. p336, Example 1.16: There is some confusion here. The paper by Oguiso and Zhang is about the case that$\varphi(n)= rk T(X)$. So replace m by n here. (Thanks to Eva Bayer-Fluckiger for pointing this out.) 19. p351, Theorem 3.13: In the list, the group$(Z/4Z)^4$should be replaced by$(Z/4Z)^2$. (Thanks to Georg Oberdieck for pointing this out.) 20. p405, l-10: At this point one should recall the notation from Section 4.4.1 that$\bar X=X\times_k\bar k$. 21. p419, Cor. 1.13, Cor. 1.15:$\hat Br_X$has the wrong font. 22. ~ Math: 1. p xi: It was pointed out to me that Broad Peak may not be after all what professionals call K3 (which probably is Gasherbrum IV). (Thanks to Boris Kruglikov for the correction.) 2. p.111: Since the$H^{1,1}$do not form a holomorphic subbundle, the map$p$is not holomorphic. One should instead consider$F^1$. (Thanks to Yagna Dutta for communicating this mistake, pointed out to her by C. Voisin.) 3. p.118, Theorem 4.1: This is not correctly stated. Borel's theorem indeed needs the assumption of the group being torsion free (and so the quotient being smooth). The argument how to reduce to this case is however correct. (Thanks to Ariyan Javanpeykar for pointing it out and explanation concerning this point.) 4. p212, Proposition 2.5: One needs to assume that$\chi(E)\ne0$(as indeed is assumed in Yoshioka's paper). In the proof, I tried to reduce to this case by twisting with one large$H$, but after that I still need to argue for all$H$. (Thanks to Arend Bayer and Nick Addington for pointing this out to me.)) 5. p246, Proposition 5.4: This is wrong as stated. One has to assume that$X\to {\mathbb P}^1$has no multiple fibre. In fact, in the second approach this was explicitly used. How exactly this enters the first approach mentioned there still eludes me. Correspondingly, (i) in Corollary 5.5 is just wrong. The map from the Tate--Safarevic group to the Weil--Chatelet group is injective, but its cokernel is$\bigoplus H_1(X_t,{\mathbb Q}/{\mathbb Z})$. See Friedman's book [185] for a discussion. Accordingly, the comment after Definition 5.11, page 250 is wrong. (Thanks to Igor Dolgachev for pointing this out to me.) 6. p252, Remark 5.15: It should be parametrizing all elliptic surfaces without multiple fibres' (due to the above). 7. p343, l-8:$f$cannot be symplectic, just apply the Lefschetz fixed point formula. (Thanks to Serge Cantat for pointing this out.) 8. p343, Section 2.5: As Serge Cantat pointed out to me, K3 surfaces that are double cover of Enriques surfaces have large automorphism group. Indeed, the automorphism group of an Enriques surface is a subgroup of finite index of the orthogonal group of its integral cohomology. 9. p384: Arend Bayer explained to me how to prove that$E^\perp$for$E\$ spherical on a K3 surface of Picard number one is not empty.

10. p412, l3: It is not true that every Brauer class in Br(k) of a field k becomes trivial after base change to some Galois extension of degree equal the index. This corresponds to the question whether every divison algebra is a crossed product. See for example the discussion page 3 of Open problems on central simple algebras' by Auel, Brussel, Garibaldi, and Vishne. (This was brought to my attention by J.-P. Serre.)