We discuss the theory of Hilbert--Samuel multiplicities: their connections with integral closure, Koszul homology, and singularities. We then focus on a long standing conjecture of Lech which states that the multiplicities do not drop under faithfully flat extensions of local rings R-->S. We survey the literature of this conjecture and various attempts to attack it. Finally, we discuss some very recent work that proves Lech's conjecture when R is standard graded, using lim Ulrich sequence and weakly lim Ulrich sequence that we introduce. Roughly speaking, these are sequences of finitely generated modules that are not necessarily Cohen--Macaulay, but asymptotically behave like Ulrich modules. We show their existence imply Lech's conjecture, and we construct weakly lim Ulrich sequence for standard graded rings of positive characteristic.
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