New lapped transforms are introduced. The LBT (lapped biothogonal transform), and HLBT (hierarchical lapped biorthogonal transform) are appropriate for image coding, and the MLBT (modulated lapped biorthogonal transform) and NMLBT (nonuniform modulated lapped biorthogonal transform) are appropriate for audio coding. The HLBT has a significantly lower computational complexity than the LOT (lapped orthogonal transform), essentially no blocking artifacts, and less ringing artifacts than the commonly-used DCT (discrete cosine transform). The LBT and HLBT have transform coding gains that are typically between 0.5 and 1.2 dB higher than that of the DCT. Image coding examples using JPEG and embedded zerotree coders demonstrate the better performance of the LBT and HLBT. The NMLBT has less ringing artifacts and better reproduction of transient sounds than the MLT< as shown in audio coding examples. Fast algorithms for both the HLBT and the NMLBT are presented.