A sequence \((x_i)_{i\in l}\) of unit vectors in a Banach space \(X\) is called a \(\mu\)-approximate \(\ell_1\) system if \(\|\sum_{i\in A}\pm x_i \|\geq |A |-\mu\) for all finite sets \(A\subset I\), where \(I=\{1,2, \dots, n\}\) or \(I=\mathbb{N}\) and \(\mu\geq 0\). The authors prove that a Banach space contains an infinite \(\mu\)-approximate \(\ell_1\) system for some \(\mu\geq 0\) if and only if \(X\) contains an asymptotically isometric copy of \(\ell_1\) or, equivalently, the dual space \(X^*\) contains an isometric copy of \(L_1[0,1]\). Concerning the problem of extracting a large subsystem that is \((1+ \varepsilon) \)-equivalent to the unit vector basis of \(\ell_1^n\), the main result is the following: Theorem. Let \(\alpha\in (0,1/4)\) and \(\mu\in (0,1)\). If \(n\) is a power of 2 then there exists a norm \(\|\cdot \|\) on \(\mathbb{R}^n\) with the following properties: (i) \((e_i)^n_{i=1}\) is a suppression 1-unconditional normalized basis of \((\mathbb{R}^n,\|\cdot\|)\). (ii) \(\|\sum^n_{i=1} \pm e_i\|\geq n-\mu\) for all choices of signs. (iii) For every \(A\subseteq \{1,\dots, n\}\), with \(|A|=1+ \lceil(3/4+ \alpha)n\rceil\), there exists a nonzero vector \(x\), with \(\text{supp} x\subseteq A\), such that \(\|x\|_1\geq (1+ {4\alpha \mu\over 3+8\alpha -4\alpha\mu}) \|x\|\). (Theorem 3.4) In the second part of this paper, the authors consider the \(f(n)\)-approximate \(\ell_1\) systems, i.e., the systems \((x_i)_{i\in I}\) such that \(\|\sum_{i\in I} \pm x_i\|\geq|A|-f(|A|)\) for all finite sets \(A\subset I\) and for all choices of sign, where \((f(n))_{n\in\mathbb{N}}\) is a strictly positive nondecreasing and unbounded sequence. Some interesting characterizations of Banach spaces which contain such systems for all \((f(n))\) are given. Also, special results are obtained in the case of hereditarily indecomposable spaces isomorphic to \(\ell_1\). A typical result is the following: Suppose that \(X\) is isomorphic to \(\ell_1\). Then, given \((f(n))\) and \(\alpha >1\), \(X\) contains a normalized basic sequence \((y_n)\) satisfying the following: (i) \((y_n)\) is an \((f(n))\)-approximate \(\ell_1\) system. (ii) \((y_n)\) is a conditional basis for its closed linear span with basis constant at most \(\alpha\). (iii) The unit vector basis of \(\ell^k_2\) \((k\geq 1)\) is uniformly equivalent to a black basis of \((y_n)\). If \(X=\ell_1\), we may also take \((y_i)\) to be a monotone basic sequence. In the last part of this paper, the authors obtain existence results concerning lacunary Haar systems in \(L_1[0,1]\) which are quasi-greedy bases for their linear span.