The author studies numerical invariants of ``curves'', i.e. one-dimensional Cohen-Macaulay rings \((A,{\mathfrak m})\), and investigates their relation to the Cohen-Macaulay property of the tangent cone of \(R\), i.e. the associated graded ring \(\text{gr}_{\mathfrak m}(A)\). The numerical invariants under consideration are the embedding dimension \(b=b(R)\), the multiplicity \(e=e(R)\), the index of regularity \(i= i(R)\), the starting degree \(s=s(R)\) of the defining ideal of \(R\) and other data derived from the Hilbert-Samuel function. In order to show the existence of curves with prescribed data the author first develops some techniques for the construction of curves. He then characterizes those triples \((b,e,i)\) for which there exists a curve \(R\): This is the case if (i) \(b=e=1\), \(i=0\), or (ii) \(2\leq b\leq e-1\) and \(r+1\leq i\leq e-1\), or (iii) \(b=e\) and \(i=1\). The next main result lists cases for which there (do not) exist curves with Cohen-Macaulay tangent cone: If \(i>e-b+1\), then \(\text{gr}_{\mathfrak m}(R)\) is never Cohen-Macaulay, and for \(r+3\leq i\leq e-b+1\) one can always find curves \(X\) and \(Y\) with Cohen-Macaulay and non-Cohen-Macaylay tangent cones, respectively. Thus there remain the cases \(i=r+1\) and \(i=r+2\). For their study also \(s\) plays a role, and the author finds some cases for which there (do not) exist curves with Cohen-Macaulay tangent cones. He also gives some conditions in terms of the Hilbert-Samuel function that are satisfied in the case of existence. The last part of the paper is devoted to curves with multiplicity at most 6.