Context. The increase in bandwidth and sensitivity of state-of-the-art radio observatories is providing a wealth of molecular data from nearby star-forming regions up to high-z galaxies. Analysing large data sets of spectral cubes requires efficient and user-friendly tools optimised for astronomers with a wide range of backgrounds. Aims. In this paper we present the detailed formalism at the core of Spectral Line Identification and Modelling (SLIM) within the MAdrid Data CUBe Analysis (MADCUBA) package and their main data-handling functionalities. These tools have been developed to visualise, analyse, and model large spectroscopic data cubes. Methods. We present the highly interactive on-the-fly visualisation and modelling tools of MADCUBA and SLIM, which includes a stand-alone spectroscopic database. The parameters stored therein are used to solve the full radiative transfer equation under local thermodynamic equilibrium (LTE). The SLIM package provides tools to generate synthetic LTE model spectra based on input physical parameters of column density, excitation temperature, velocity, line width, and source size. It also provides an automatic fitting algorithm to obtain the physical parameters (with their associated errors) better fitting the observations. Synthetic spectra can be overlayed in the data cubes/spectra to ease the task of multi-molecular line identification and modelling. Results. We present the Java-based MADCUBA and its internal module SLIM packages which provide all the necessary tools for manipulation and analysis of spectroscopic data cubes. We describe in detail the spectroscopic fitting equations and make use of this tool to explore the breaking conditions and implicit errors of commonly used approximations in the literature. Conclusions. Easy-to-use tools like MADCUBA allow users to derive physical information from spectroscopic data without the need for simple approximations. The SLIM tool allows the full radiative transfer equation to be used, and to interactively explore the space of physical parameters and associated uncertainties from observational data.