We present dielectric measurements of the colossal dielectric constant material CaCu 3 Ti 4 O 12 extending up to 1.3 GHz also covering so far only rarely investigated single crystalline samples. Special emphasis is put on the second relaxation reported in several works on polycrystals, which we detect also in single crystals. For polycrystalline samples we provide a recipe to achieve values of the dielectric constant as high as in single crystals.Among the vast number of papers on the extremely high ("colossal") dielectric constants (ε') found in CaCu 3 Ti 4 O 12 (CCTO) there are at least ten so-called "highly-cited" papers 1,2,3,4,5,6,7,8,9,10 . This demonstrates the tremendous interest in new high-ε' materials, which are prerequisite for further advances in the development of capacitive electronic elements. It soon became clear that the colossal ε' in CCTO must have a non-intrinsic origin 4,5,6,7,8,9,11,12,13 . Nowadays usually the results are interpreted within an "internal barrier layer capacitor" (IBLC) picture: Polarization effects at insulating grain boundaries between semiconducting grains or other internal barriers generate non-intrinsic colossal values of ε', accompanied by a strong Maxwell-Wagner (MW) relaxation mode. As an alternative, a "surface barrier layer capacitor" (SBLC) picture was proposed, assuming, e.g., the formation of Schottky diodes at the contact-bulk interfaces 8,9,14 . Nearly all experimental evidence for a non-intrinsic mechanism so far is based on measurements of ceramic samples (e.g., refs. 4,5,6,7,9,11,12,13). However, extremely high values of ε' of the order of 10 5 were observed particularly in CCTO single crystals (SCs) 3 . With only few exceptions 11 , measurements of polycrystals (PCs) reveal much lower values in the range of 10 3 -10 4 . There are no grain boundaries in SCs and thus there were speculations about other internal boundaries as, e.g., twinboundaries 4,6,15,16 . However, if there is any kind of planar defects in SCs generating strong relaxations, should they not contribute to a separate relaxational response in PCs, where grains can reach sizes up to 100 µm, too? Interestingly, it seems clear now that indeed there is a second relaxation in CCTO PCs 9,12,14 and it even was already observed in one of the earliest reports on CCTO 2 .It leads to even larger ε' values than the well-known main relaxation. Thus one could speculate that one of the two relaxations is due to grain boundaries and the other due to planar defects within the grains. Alternatively, one relaxation could be due to an IBLC and the other to a SBLC mechanism 9 . In the present work we aim to elucidate differences and similarities in SC and PC behavior, address the question of the second relaxation, and try to help solving the IBLC/SBLC controversy. For this purpose, we performed measurements on various samples including single-crystals, which to our knowledge so far only were investigated in two works 3,16 . The spectra cover up to nine frequency decades and extend up to the technically relev...